Piezoelectricity and Topological Quantum Phase Transitions in Two-Dimensional Spin-Orbit Coupled Crystals with Time-Reversal Symmetry
PPiezoelectricity and Topological Quantum Phase Transitions in Two-DimensionalSpin-Orbit Coupled Crystals with Time-Reversal Symmetry
Jiabin Yu and Chao-Xing Liu ∗ Department of Physics, the Pennsylvania State University, University Park, PA, 16802
Finding new physical responses that signal topological quantum phase transitions is of boththeoretical and experimental importance. Here, we demonstrate that the piezoelectric responsecan change discontinuously across a topological quantum phase transition in two-dimensional time-reversal invariant systems with spin-orbit coupling, thus serving as a direct probe of the transition.We study all gap closing cases for all 7 plane groups that allow non-vanishing piezoelectricity and findthat any gap closing with 1 fine-tuning parameter between two gapped states changes either the Z invariant or the locally stable valley Chern number. The jump of the piezoelectric response is foundto exist for all these transitions, and we propose the HgTe/CdTe quantum well and BaMnSb astwo potential experimental platforms. Our work provides a general theoretical framework to classifytopological quantum phase transitions and reveals their ubiquitous relation to the piezoelectricresponse. CONTENTS
I. Introduction 2II. Results 3A. PET jump across a Direct QSH-NI TQPT 3B. Classification of Direct 2D TQPTs and PET jumps for 7 PGs 5C. HgTe/CdTe Quantum Well 7D. Layered Material BaMnSb p p m , c m
1, and p g p p m and p m p p m c m p g U ∈ G but T / ∈ G UT ∈ G but T / ∈ G G p C ∈ G and T / ∈ G G p m and PG p m ∗ [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y a. Scenario (i): TRIM 19b. Scenario (ii): U ∈ G and T / ∈ G UT ∈ G and T / ∈ G G C p p m c m
1, and p g U ∈ G but T / ∈ G UT ∈ G but T / ∈ G p C ∈ G and T / ∈ G p m and p m UT ∈ G and T / ∈ G d -induced PET jump for E = 0 262. E -induced PET jump for fixed d I. INTRODUCTION
The discovery of topological phases and topological phase transitions has revolutionized our understanding ofquantum states of matter and quantum phase transitions (1–3). Two topologically distinct gapped phases cannot beadiabatically connected; if the system continuously evolves from one phase to the other, a topological quantum phasetransition (TQPT) with the energy gap closing (GC) must occur. A direct way to probe such TQPTs is to detectthe discontinuous change of certain physical response functions. Celebrated examples include the jump of the Hallconductance across the plateau transition in the integer quantum Hall system (4, 5), the jump of the two-terminalconductance across the TQPT between the quantum spin Hall (QSH) state and normal insulator (NI) state in atwo-dimensional (2D) time-reversal (TR) invariant system (6), and the jump of the magnetoelectric coefficient acrossthe TQPT between the strong topological insulator phase and NI phase in a 3D TR invariant system (7–10). Thephysical responses in all these examples are induced by the electromagnetic field. A natural question then arises: canwe detect TQPTs with other types of perturbation?Here we theoretically answer this question in the affirmative: the discontinuous change of the piezoelectric responseis a ubiquitous and direct signature of 2D TQPTs. The piezoelectric effect, the electric charge response induced bythe applied strain, is characterized by the piezoelectric tensor (PET) to the leading order. PET was originally definedto relate the change of the the charge polarization P with the infinitesimal homogeneous strain, which reads (11) γ ijk = ∂P i ∂u jk (cid:12)(cid:12)(cid:12)(cid:12) u jk → , (1)where u ij = ( ∂ x i u j + ∂ x j u i ) / u is the displacement at x . The modern theory of polariza-tion (12–14) later identified the above definition as improper (15) due to the ambiguity of P in crystals, while theproper definition adds the adiabatic time dependence to u jk and relates it to the bulk current density J i that canchange the surface charge: γ ijk = ∂J i ∂ ˙ u jk (cid:12)(cid:12)(cid:12)(cid:12) u jk , ˙ u jk → . (2)With Eq. (2), the PET of an insulating crystal has been derived as (15, 16) γ ijk = − e (cid:90) d k (2 π ) (cid:88) n F nk i ,u jk (cid:12)(cid:12)(cid:12) u jk → , (3)where the integral is over the entire first Brillouin zone (1BZ), and n ranges over all occupied bands. The F nk i ,u jk term has a Berry-curvature-like expression F nk i ,u jk = ( − i) (cid:2) (cid:104) ∂ k i ϕ n, k | ∂ u jk ϕ n, k (cid:105) − ( k i ↔ u jk ) (cid:3) (4)with | ϕ n, k (cid:105) the periodic part of the Bloch state in the presence of the strain. (See the Methods for more details.) Theexpression indicates an extreme similarity between Eq. (3) and the expression for the Chern number (CN) (5). It isthis similarity that motivates us to study the relation between the PET and the TQPT.Despite the similarity, the topology connected to the PET is essentially different from the CN, since the PET canexist in TR invariant systems whose CNs always vanish. We, in this work, study the piezoelectric response of 2DTR invariant systems in the presence of the significant spin-orbit coupling (SOC) and demonstrate the jump of allsymmetry-allowed PET components across the TQPT. In particular, we focus on the 7 out of the 17 plane groups(PGs) that allow non-vanishing PET components (17, 18), including p p m c m p g p p m
1, and p m .The two-fold rotation C (with the axis perpendicular to the 2D plane) or the 2D inversion restricts the PET tozero in the other 10 PGs (19), according to γ ijk = (cid:80) i (cid:48) j (cid:48) k (cid:48) R ii (cid:48) R jj (cid:48) R kk (cid:48) γ i (cid:48) j (cid:48) k (cid:48) for any O (2) symmetry R of the 2Dmaterial. Through a systematic study, we find that any GC between two gapped states that only requires 1 fine-tuningparameter is a TQPT in the sense that it changes either the Z index (1, 2) or the valley CN (20). Although thechange of the valley CN is locally stable (21), we still treat the corresponding GC as a TQPT, since the two statescannot be adiabatically connected when the valley is well defined. All the TQPTs contain no stable gapless phase inbetween two gapped phases, and thereby we refer to them as the direct TQPTs. All PET components that are allowedby the crystalline symmetry exhibit discontinuous changes across any of the direct TQPTs, showing the ubiquitousconnection. Interestingly, when the gap closes at momenta that are not TR invariant, the strain tensor u ij acts as apseudo-gauge field (22) at the TQPT, making the PET jump directly proportional to the change of the Z index orthe valley CN.Our work presents a general framework for the PET jump across the TQPT in 2D TR invariant systems with SOC.The relation between the PET and the valley CN in the low-energy effective model has been studied in graphene witha staggered potential (23), h-BN (24, 25), and monolayer transition metal dichalcogenides (TMDs) XY for X=Mo/Wand Y=S/Se (25). However, these early works have not pointed out that it is the PET jump (well described within thelow-energy effective model) that is the experimental signature directly related to the TQPT, while the PET itself atfixed parameters might contain the non-topological background given by high-energy bands. Moreover, these works,unlike our systematic study, only considered one specific plane group ( p m
1) around one specific type of momenta(
K, K (cid:48) ). The relation between the PET and the Z index were not explored either. Besides, graphene and h-BN haveneglectable SOC, and the TMDs have a large gap, making them not suitable for realizing TQPT. We thereby proposetwo realistic material systems, the HgTe/CdTe quantum well (QW) and the layered material BaMnSb , as potentialexperimental platforms. The Z TQPT and PET jump can be achieved by varying the thickness or the gate voltagesin the HgTe/CdTe QW or by tuning lattice distortion in BaMnSb . II. RESULTSA. PET jump across a Direct QSH-NI TQPT
We start from a simple example of the TQPT discussed in Ref. (26). They (in the example of our interest) consideredthe case with no crystalline symmetries other than the lattice translation (PG p
1) and focused on the GC at twomomenta ± k that are not TR invariant momenta (TRIM), as labeled by red crosses in Fig. 1(a). The low-energyeffective theory for the electron around k can be described by the Hamiltonian of a 2D massive Dirac fermion (26) h + , ( q ) = E ( q ) σ + v x q σ x + v y q σ y + mσ z , (5)where q = k − k , m is the tuning parameter for the TQPT, and σ ’s are Pauli matrices. In the above Hamiltonian, theunitary transformation on the bases and the scaling/rotation of q are performed for the simplicity of the Hamiltonian;the latter generally makes q , q along two non-orthogonal directions. (See Appendix C for details.) The effectiveHamiltonian at − k is related to h + , by the TR symmetry. After choosing appropriate bases at − k , the TRsymmetry can be represented as T ˙=i σ y K with K the complex conjugate, leading to h − , ( q ) = E ( − q ) σ + v x q σ x + v y q σ y − mσ z . (6)According to Ref. (26), the TQPT between the QSH insulator and the NI (distinguished by the Z index) occurswhen the mass m in h ± , ( q ) changes its sign. The argument used to determine change of the Z index was presentedin Ref. (27) and is discussed below for integrity. Since there is no inversion symmetry in PG p
1, the Z index can bedetermined from the CN of the contracted half first Brillouin zone (1BZ), where the half 1BZ is chosen such that itsKramers’ partner covers the other half. Specifically, the Z index is changed (unchanged) by the GC if the CN ofthe contracted half 1BZ changes by an odd (even) integer. Without loss of generality, let us choose the half 1BZ tocontain k , as shown in Fig. 1(a). Since h + , is a 2D gapped Dirac Hamiltonian, the CN of the contracted half 1BZchanges by ∆ N + = − sgn( v x v y ) as m increases from 0 − to 0 + , featuring a direct QSH-NI TQPT as v x v y is typicallynonzero.We next discuss the piezoelectric effect in this simple effective model. To do so, we need to introduce the electron-strain coupling around ± k based on the TR symmetry: h ± , ( u ) = ξ ,ij σ u ij ± ξ a (cid:48) ,ij σ a (cid:48) u ij , (7)where the duplicated indexes, including a (cid:48) = x, y, z and i, j = 1 ,
2, are summed over henceforth unless specifiedotherwise. ξ ’s are the material-dependent coupling constants between the low-energy electrons and the strain tensor,which obey ξ a,ij = ξ a,ji with a = 0 , x, y, z owing to u ij = u ji and are related to the electron-phonon coupling (28).The full form of the effective Hamiltonian is then given by h ± ( q , u ) = h ± , ( q ) + h ± , ( u ) . (8)To use Eq. (3), we simplify Eq. (8) by neglecting the E term, which has no influence on the piezoelectric response ofinsulators (see Appendix A). When ξ x,ij = ξ y,ij = 0, the Hamiltonian h ± has effective inversion symmetry within eachvalley, σ z h ± ( − q , u ) σ z = h ± ( q , u ), which forbids the piezoelectric effect. Thus, ξ ,ij and ξ z,ij terms cannot contributeto the PET, and neglecting them leads to a further simplified version of Eq. (8): h ± ( q , u ) = [ v x ( q ± A pse )] σ x + [ v y ( q ± A pse )] σ y ± mσ z , (9)where A pse = ξ x,ij u ij /v x and A pse = ξ y,ij u ij /v y . The above form suggests that the remaining strain terms, ξ x,ij and ξ y,ij , serve as the pseudo-gauge field A psei that has opposite signs for two valleys ± k (10, 22, 25, 29). As thestrain tensor only exists in the form of q i ± A psei , the derivative with respect to u ij in Eq. (3) can be replaced by thederivative with respect to the momentum as ∂ u ij | ϕ ± , q (cid:105) = ∂A psei (cid:48) ∂u ij ∂ A psei (cid:48) | ϕ ± , q (cid:105) = ± ∂A psei (cid:48) ∂u ij ∂ q i (cid:48) | ϕ ± , q (cid:105) , (10)where ϕ ± are the occupied bands of h ± . Substituting the above equation into Eq. (3) leads to γ eff ij = − e (cid:90) d q (2 π ) (cid:88) α = ± αF α ( q ) ∂A pse ∂u ij γ eff ij = e (cid:90) d q (2 π ) (cid:88) α = ± αF α ( q ) ∂A pse ∂u ij , (11)where F ± ( q ) is the conventional Berry curvature of the occupied band of h ± ( q , eff means thatwe neglect the contribution from bands beyond the effective model Eq. (8), indicating that the above equation is notthe complete PET. Nevertheless, it can accurately give the PET change across the TQPT since high-energy bandsexperience an adiabatic deformation and the corresponding background PET contribution should remain unchangedat the transition ( m = 0). As m varies from 0 − to 0 + , Eq. (11) gives the change of PET ∆ γ ijk as∆ γ ij = − e ∆ N + π ξ y,ij v y ∆ γ ij = e ∆ N + π ξ x,ij v x . (12)The PET jump shown in the above equation is nonzero since v x v y and the electron-strain coupling ξ ’s are typicallynon-zero. We thus conclude that for p Z index occurs across the TQPT, when the gap closes not at TRIM.The PET jump can be physically understood based on Eq. (2). Let first focus on one GC momentum, say k . Sincethe strain tensor couples to the electron in the way similar to the U (1) gauge field as shown in Eq. (8), ˙ u jk shouldact like a electric field on the electron. According to Eq. (2), γ ijk should then behave like the Hall conductance,whose jump is proportional to the change of CN ∆ N + . Now we include the other GC momentum − k . Unlike theactual U (1) gauge field, the pseudo-gauge field given by the strain couples oppositely to the electron at the two GCmomenta (Eq. (8)). The opposite signs of the coupling can cancel the opposite signs of the Berry curvature, and thus,in contrast to the actual Hall conductance, the contributions to γ ijk from ± k add up to a nonzero value instead ofcanceling each other, leading to the non-zero topological jump in Eq. (B2). (a) (b) (c) Γ𝑀 𝐾 𝑝3 : (i) 𝑀 𝑀 𝐾′ 𝑝3 : (ii) 𝑝3 : (iii) Γ𝑀 𝑀𝑀 𝐾𝐾′ 𝑝31𝑚 : (i) 𝑝31𝑚 : (iii) Γ 𝑀 𝐾𝑀 𝑀𝐾′ 𝑝3𝑚1 : (i) 𝑝3𝑚1 : (iii) 𝑝3𝑚1 : (iii) (d) (e)(f) (g) (h) (i) (j) 𝑝31𝑚 : (iv) 𝑝3𝑚1 : (iv) (k) (l) (m) (n) (o)
Γ 𝑀𝑌 𝑋 Γ 𝑝1𝑚1, 𝑝1𝑔1 : (iii) 𝑐1𝑚1 : (iii) 𝑝1𝑚1, 𝑝1𝑔1 : (iv) 𝑐1𝑚1 : (iv)
Γ 𝑀𝑌 𝑋 𝑝1 : (ii) Fig 1: GC cases with 1 fine-tuning parameter.
The figure shows the GC cases between insulating states with1 fine-tuning parameter for all 7 PGs with non-vanishing PET. The red cross labels the GC momenta, the light bluebackground indicates the 1BZ, and the light red part in (a) indicates the half 1BZ. The black and orange dashedlines label the momenta invariant under the mirror/glide symmetry and the combination of mirror/glide and TRsymmetries, respectively. The figures are first grouped according to the PGs and then ordered based on the GCscenarios listed in Tab. 1, whose labels are next to the names of the PGs.
B. Classification of Direct 2D TQPTs and PET jumps for 7 PGs
The above section discusses an example of 2D QSH-NI TQPT for the p p p m c m p g p p m , and p m
1. The main results are summarized in Fig. 1and Tab. 1, as discussed below. The other 10 PGs ( p p mm , p mg , p gg , c mm , p p mm , p gm , p
6, and p mm )have vanishing PET due to the existence of inversion symmetry or C rotation symmetry, and are briefly discussedin Appendix B.TQPTs in different PGs can be analyzed in the following three steps. In the first step, we classify the GC based onthe GC momenta and the symmetry property of the bands involved in the GC. To do so, we define the group G for aGC momentum k such that G contains all symmetry operations that leave k invariant (including the little groupof k and TR-related operations). We start with a coarse classification based on G , which leads to 2 scenarios for p p
3, and 4 scenarios for p m c m p g p m , and p m
1, as listed in Tab. 1 and the Methods. Toillustrate this classification, we consider the p T ∈ G ), i.e. the Γ point or three M points in Fig. 1(f). In scenario (ii), the GCoccurs simultaneously at K and K (cid:48) where G contains C but no T (Fig. 1(g)). In scenario (iii), the GC occurs atsix generic momenta ( G only contains lattice translations) that are related by C rotation and TR (Fig. 1(h)). Theclassification of GC momenta is coarse here since G can still vary within one scenario. For example, in scenario (i) of p G at Γ contains C while G at M does not. Moreover, even at a certain GC momentum with a certain G , thesymmetry properties of bands involved in the GC may vary. For example, at K in scenario (ii) of p
3, the gap mayclose between two states with the same or different C eigenvalues. Therefore, we further refine our classification bytaking these subtleties into consideration and classify each GC scenario into finer GC cases.In the second step, for each GC case, we construct a symmetry-allowed low-energy effective Hamiltonian that wellcaptures the GC and count the number of fine-tuning parameters. Since G and the symmetry properties of the bandsinvolved in the GC are fixed in one GC case, the form of the effective Hamiltonian can be unambiguously determined.(See details in Appendix B and C.) After obtaining the effective Hamiltonian, we can count the number of fine-tuningparameters required for each GC and select out all GC cases that require only 1 fine-tuning parameter (or equivalentlyhas codimension 1), as shown in Fig. 1. Only these cases can be direct TQPTs between two gapped phases, sinceany two gapped states in the parameter space are adiabatically connected if 2 or more fine-tuning parameters arerequired to close the gap, and 0 codimension means there is a stable gapless phase in between two gapped phases.Our analysis shows that all GC cases in scenarios (i) for p
1, (i) and (ii) for p m c m
1, and p g
1, and (ii) for p m p m need 0 fine-tuning parameter or more than 1 fine-tuning parameters and thus cannot correspond to thedirect TQPTs, while codimension-1 GC cases can exist in all other scenarios.In the third and final step, we demonstrate the topological nature of all the codimension-1 GC cases by evaluatingthe change of certain topological invariants and derive the corresponding PET jump. As shown in Tab. 1, the Z index is changed in all codimension-1 GC cases of scenarios (ii) for p
1, (iii) for p m c m
1, and p g
1, (i)-(iii) for p
3, and (i) and (iii) for p m p m , while the valley CN is changed for all codimension-1 GC cases of thescenarios (iv) for p m c m p g p m
1, and p m . We would like to emphasize that although valley CN itself isin general not quantized in a gapped phase, the change of valley CN across a gap closing is quantized and has physicalconsequence (46). (See the Methods for more details.) According to Fig. 1, the Z cases either close the gap at TRIMor have an odd number of Dirac cones in half 1BZ, while all the valley CN cases (Fig. 1(d-e) and Fig. 1(n-o)) have aneven number of Dirac cones in half 1BZ, forbidding the change of the Z index. Nevertheless, no matter which type,they all lead to discontinuous changes of the symmetry-allowed PET components. (See detailed calculation of PETin Appendix B.)In sum, we conclude that for all 7 PGs with non-vanishing PET, all the GC cases between two gapped phaseswith 1 fine-tuning parameter are direct TQPTs that change either Z index or valley CN, and they all induce thediscontinuous change of the symmetry-allowed PET components. Based on these results, we propose the followingcriteria to find realistic systems to test our theoretical predictions: (i) whether it breaks the 2D inversion or two-foldrotation with axis perpendicular to the 2D plane, (ii) whether it has significant SOC, and (iii) whether there is atunable way to realize the GC. Applying these conditions to the existing material systems for 2D TQPT, we find tworealistic material systems, namely the HgTe/CdTe QW and the layered material BaMnSb , which are studied in thefollowing. PGs p p m c m p g p p m p m Scenario (i) (ii) (i) (ii) (iii) (iv) (i) (ii) (iii) (i) (ii) (iii) (iv)Codim-1 GC × (a) × × (b-c) (d-e) (f) (g) (h) (i-j) × (k-m) (n-o)Topo. Inv. N/A Z N/A N/A Z VCN Z Z Z Z N/A Z VCNPET Jump N/A (cid:88)
N/A N/A (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88)
N/A (cid:88) (cid:88)
Table 1: Summary for all 7 PGs with non-vanishing PET.
The scenarios are classified by the symmetriesthat leave the GC momenta invariant, as shown in the Methods. Codim-1 GC means the GC cases with 1fine-tuning parameter or codimension 1. If at least one GC case between gapped states with 1 fine-tuning parameterexists in the corresponding scenario, the subfigures in Fig. 1 that illustrate the GC momenta are referred to;otherwise, we fill in a × . Topo. Inv. labels the topological invariant changed by the GC, Z means the Z index,and VCN means the corresponding case changes the valley CN when the valley is well-defined. C. HgTe/CdTe Quantum Well
It has been demonstrated (6, 47) that the TQPT between the QSH insulator and NI phases in the HgTe/CdTeQW can be achieved by tuning the HgTe thickness d . Tuning applied electric field E was theoretically predictedas an alternative way to achieve TQPT (48, 49), making the system an ideal platform to study the PET jump atTQPTs. Here, the stacking direction of the QW is chosen to be (111) instead of the well-studied (001) direction (50),since the latter would allow a two-fold rotation that forbids PET. Without the applied electric field, the (111)QW has the TR symmetry and the C v symmetries (generated by three-fold rotation along (111) and the mirrorperpendicular to (¯110)); adding electric field along (111) does not change the symmetry properties. We should thenexpect one independent symmetry-allowed PET component γ similar to Eq. (B14) in the Methods, where 2 labelsthe direction (11¯2).The electronic band structure of the (111) QW can be described by the 6-band Kane model with the bases ( | Γ , ± (cid:105) , | Γ , ± (cid:105) , | Γ , ± (cid:105) ). The electric field E along (111) can be introduced by adding a linear electric potential that isindependent of orbitals and spins. In this electron Hamiltonian, there are two inversion-breaking (IB) effects, theinherent IB effect in the Kane model and the applied electric field, and we neglect the former for simplicity. Notethat such approximation does not lead to vanishing PET even for E = 0 because the IB electron-strain coupling willbe kept.We first discuss the inversion-invariant E = 0 case and focus on the PET jump induced by varying the width d .In this case, there are two double degenerate bands closest to the Fermi energy, namely | E , ±(cid:105) and | H , ±(cid:105) bandswith opposite parities. With the method proposed in Ref. (6), we find that the gap between two bands closes at theΓ point around d = 65˚A as shown in Fig. 2(a). The GC must be a Z TQPT owing to the opposite parities of thetwo bands, and it belongs to scenario (i) of p m /p m discussed in Tab. 1 and the Methods. We further include theelectron-strain coupling, and numerically plot the independent PET component γ as the function of the width inFig. 2(b), which shows a jump around d = 65˚A. (See Appendix E.)Next we study the TQPT induced by the applied electric field. In order to realize the GC at a nonzero value ofthe electric field, we fix the width of the QW at d = 62˚A, away from 65˚A. After adding the linear electric potentialalong (111) in the 6-band Kane model, we numerically find that the GC at Γ point happens at E ≈ . − , asshown in Fig. 2(c). Such GC belongs to scenario (i) of p m /p m and is still a Z TQPT since the extra IB termcannot influence the Z topology change. The PET component γ is numerically shown in Fig. 2(d), showing thejump across the TQPT. The PET jump in Fig. 2(b) and (d) has the order 10 ∼ − , and thus is possible tobe probed by the current experimental technique (51). D. Layered Material BaMnSb BaMnSb is a 3D layered material that consists of Ba-Sb layers and Mn-Sb layers, which are stacked alternativelyalong the (001) direction (or equivalently z direction). The electrons in p x and p y orbitals of Sb atoms in the Ba-Sblayers account for the transport of the material. Owing to the insulating Mn-Sb layers, the tunneling along the z direction among different Ba-Sb layers is much weaker than the in-plane hopping terms, and thus BaMnSb can betreated as a quasi-2D material (52). Therefore, we can only consider one Ba-Sb layer, whose structure is shown inFig. 3(a). Owing to the zig-zag distortion of the Sb atoms (solid lines in Fig. 3(a)), the symmetry group that capturesthe main physics is spanned by the TR symmetry T and two mirror operations m y and m z that are perpendicular (a) (b) 𝑑/Å 𝛾 e Å −1 𝐸 eV (d)(c) 𝛾 e Å −1 E1 H1 𝑚meV
ℰ/(V/Å) 𝑑 = 62 Å
ℰ/(V/Å) 𝑑/Å
Fig 2: HgTe/CdTe QW.
This figure shows the energy dispersion and the PET of the HgTe QW with thestacking direction (111). In ( a ), the lower panel shows the energy of E1 (blue) and H1 (red) bands at Γ point as afunction of the width d , and the upper panel shows the energy dispersion at d = 60 , , d ≈ d = 63˚A reported inRef. (47) for the (001) stacking direction owing to the anisotropy effect. ( b ) shows the PET component γ as afunction of d . In ( c ), the lower panel plots gap m as a function of the electric field E with d = 62˚A, showing that thegap closes at E ≈ . − . The upper panel of (c) demonstrates the energy dispersion at E = 0 . , . , . − from left to right, respectively. ( d ) shows the PET component γ as a function of E .to y and z axes, respectively. The mirror symmetry m z does nothing but guarantee the z-component of the spin tobe a good quantum number, allowing us to view the system as a spin-conserved TR-invariant 2D system with PG p m
1. Slightly different from the demonstration in the Methods, the mirror here is perpendicular to y instead of x , and thereby PG p m γ yyy = γ yxx = γ xyx = γ xxy = 0 and leaves the other four components assymmetry-allowed.To describe this system, a tight-binding model with p x and p y orbitals of Sb atoms was constructed in Ref. (52)based on the first-principle calculation, and the form of the model is reviewed in Appendix F for integrity. Thismodel qualitatively captures all the main features of the electronic band structure of BaMnSb . The key parameterof the model is the distortion parameter α that describes the zig-zag distortion of the Sb atoms. When α is tunedto a critical value α c ≈ .
86, the gap of the system closes at two valleys K ± = ( π, ± k y ) near X along X − M inthe BZ, as shown in Fig. 3(b). This GC results in a TQPT between the QSH state and the NI state in one Ba-Sblayer, as confirmed by the direct calculation of Z index (Fig. 3(c)) according to expression in Ref. (53). Since the twoGC momenta are invariant under T m y , this GC case satisfies the definition of scenario (iii) for p m
1. We furthernumerically verify the PET jump induced by the GC with the tight-binding model. The jump of the symmetry-allowedPET components is found at the TQPT around α = α c in Fig. 3(d), while the components forbidden by the symmetrystay zero. According to Fig. 3(d), both the jump and background are of the same order of magnitude, 0.1 e˚A − for γ yxy,yyx and 0.01 e˚A − for γ xxx,xyy , indicating that the jump is experimentally measurable. The Z topology changeand the PET jump can also be analytically verified based on the effective model discussed in Appendix F. III. DISCUSSION
In conclusion, we demonstrate that for all PGs that allow nonvanishing PET, the piezoelectric response has a dis-continuous change across any TQPT in 2D TR invariant systems with significant SOC. Potential material realizationsinclude the HgTe/CdTe quantum well and the layered material BaMnSb .The early study on MoS has demonstrated that the values of the PET obtained from the effective model might be(though not always) quite close to those from the first principles calculations (25). Therefore, although our theory isbased on the effective Hamiltonian, the predicted jump of the PET is quite likely to be significant and even the signchange of PET, such as Fig. 2(b) and (d) for the HgTe case, might exist in realistic materials. The evaluation of the (a) 𝑍 𝛼 (d) 𝐸eV 𝑘 𝑦 𝑎 𝑦 𝛼 = 0.7𝛼 = 1𝛼 = 0.86𝛼𝛾 𝑥𝑥𝑥 𝛾 𝑥𝑦𝑦 𝛾 𝑦𝑥𝑦 , 𝛾 𝑦𝑦𝑥 (b) (c) SbBa Fig 3: Layered Material BaMnSb . ( a ) illustrates the Ba-Sb layer, where each dashed circle stands for theprojection of two Ba atoms onto the Sb layer and the solid dots are Sb atoms. The solid lines connecting Sb atomsindicate the zig-zag distortion, and the red dashed box marks the unit cell with 1 and 2 labeling the two Sb atoms.( b ) The band structure of the TB model for BaMnSb along M − X − M for α = 0 .
86 (red), α = 1 (orange), and α = 0 . X is at k y = 0. ( c ) and ( d ) plot the Z index and PET components obtainedfrom the TB model as a function of α , respectively. In (d), the PET components are in the unit e˚A − , the graydashed line is at α = 0 .
86, and the inset is the zoom-in version of the boxed region.PET from the first principles calculations is left for the future works.Although we only focus on two realistic material systems in this work, the theory can be directly applied to othermaterial systems. For example, the calculations for the HgTe/CdTe QW are also applicable to InAs/GaSb QWs,which share the same model (54). The QSH effect has also been observed in the monolayer 1T’-WTe (55–57), butits inversion symmetry (58) forbids the piezoelectric effect. Therefore, a significant inversion breaking effect fromthe environment (such as substrate) is required to test our prediction in this system. While the SOC strength ingraphene is small, it has been shown that the bilayer graphene sandwiched by TMDs has enhanced SOC and serves asa platform to observe TQPT (59, 60), where the PET jump is likely to exist. The piezoelectric effect has been observedin several 2D material systems (51, 61, 62), and therefore, the material systems and the experimental technique forthe observation of the PET jump are both available. Since the PET jump is directly related to the TQPT, it furtherprovides a new experimental approach to extract the critical exponents and universality behaviors of the TQPT,which can only be analyzed through transport measurements nowadays.This work only focuses on 2D TR invariant systems with SOC, and the generalization to systems without SOC,without TR symmetry, or in 3D is left for the future. Despite the similarity between Eq. (3) and the expression of CN,the generalization to TR-breaking systems with non-zero CNs requires caution, due to the change of the definition ofpolarization (63). Another interesting question is whether the PET jump exists across the transition between statesof different higher-order (64–67) or fragile topology (34, 68). We notice that although the dynamical piezoelectriceffect may exist in metallic systems (69), its description is different from Eq. (3). It is thus intriguing to ask how thedynamical PET behaves across the transitions between insulating and semimetal phases.0 IV. METHODSA. Expression for the PET
According to Ref. (15 and 63), the expression for the PET of insulators, Eq. (3), is derived for systems with zeroCNs and within the clamped-ion approximation where ions exactly follow the homogeneous deformation and thuscannot contribute to the PET. Even though the ion contribution might be non-zero in reality, the approximation isstill legitimate in our study of PET jump since the ion contribution varies continuously across the GC of electronicbands.Eq. (3) involves the derivative of the periodic part of the Bloch state | ϕ n, k (cid:105) with respect to the strain tensor u jk . | ϕ n, k (cid:105) can always be expressed as | ϕ n, k (cid:105) = (cid:80) G f n, k , G | G (cid:105) with G the reciprocal lattice vector, and the derivative infact means | ∂ u jk ϕ n, k (cid:105) ≡ (cid:80) G ( ∂ u jk f n, k , G ) | G (cid:105) (15). In this way, the ill-defined ∂ u ij | G (cid:105) is avoided, despite that | G (cid:105) isnot continuous as changing the strain. If replacing the | ∂ u jk ϕ n, k (cid:105) in Eq. (3) by a momentum derivative | ∂ k j ϕ n, k (cid:105) with j different from i , the PET expression transforms into − eC(cid:15) ij / (2 π ), where (cid:15) ij = − (cid:15) ji , (cid:15) xy = 1, and C is the Chernnumber of the 2D insulator (5) C = (cid:90) d k π (cid:88) n F nk x ,k y . (13)This reveals the similarity between the PET expression and the expression of the CN. B. PG p For p
1, no special constraints are imposed on the PET. There are two GC scenarios for the PG p • (i) gap closes at TRIM ( T ∈ G ), • (ii) gap closes not at TRIM ( T / ∈ G ).In scenario (ii), G contains no symmetries other than the lattice translation, which we refer to as the trivial G . C. PGs p m , c m , and p g All three PGs, p m , c m
1, and p g
1, are generated by a mirror-related symmetry U and the lattice translation. U is a mirror operation for p m /c m p g
1. The difference between p m c m x , labelled as m x or g x ,respectively. The glide operation is thus denoted as g x = { m x | } , where 0 represents the translation by half theprimitive lattice vector along y . The U symmetry in these three PGs requires γ ijk = ( − i ( − j ( − k γ ijk (14)with ( − x = − − y = 1, resulting that γ xxx = γ xyy = γ yxy = γ yyx = 0 while γ xxy , γ xyx , γ yxx , γ yyy are allowedto be nonzero. For the symmetry analysis here, the PET behaves the same under the glide and mirror operationssince u ij is considered in the continuum limit. Based on G , we obtain in total 4 GC scenarios for these three PGs: • (i) the GC at TRIM ( G contains T ), • (ii) G contains U but not T , • (iii) G contains UT but not T , • (iv) G is trivial.1 D. PG p PG p C and the lattice translation. Owing to C , the PET satisfies the followingrelation γ ijk = (cid:88) i (cid:48) j (cid:48) k (cid:48) [ R ( C )] ii (cid:48) [ R ( C )] jj (cid:48) [ R ( C )] kk (cid:48) γ i (cid:48) j (cid:48) k (cid:48) , (15)where R ( C ) = − − √ √ − . (16)Solving the above equation gives two independent components γ xxx and γ yyy as γ yxy = γ yyx = γ xyy = − γ xxx γ xxy = γ xyx = γ yxx = − γ yyy . (17)Again, we classify the GC for p G , resulting in three different scenarios: • (i) G contains T , • (ii) G contains C but not T , • (iii) G is trivial.Here we do not have a scenario for G containing C T but no T , since ( C T ) is equivalent to T . E. PGs p m and p m Both PGs p m and p m C , and a mirror symmetrywhich we choose to be m x without loss of generality. The difference between the two PGs lies on the direction of themirror line relative to the primitive lattice vector: the mirror line is parallel or perpendicular to one primitive latticevector for p m or p m
1, respectively. C and m x span the point group C v , which makes the PET satisfy Eq. (14)and Eq. (15). As a result, we have γ xxx = γ xyy = γ yxy = γ yyx = 0 γ xyx = γ xxy = γ yxx = − γ yyy (18)for the PET, and thus γ yyy serves as the only independent symmetry-allowed PET component. We classify the GCscenarios into 4 types according to G : • (i) G contains T , • (ii) G contains at least one of the three mirror symmetry operations in C v (again labeled as U = m x , C m x ,or C m x ) but no T , • (iii) G contains UT but no T , • (iv) G is trivial. F. Valley CN
In all the valley CN cases (Fig. 1(d,e,n,o)), the GC points locate at generic positions in the 1BZ. The valleys can bephysically defined as the positions where the Berry curvature diverges as the gap approaches to zero. The positions ofthe Berry curvature peaks around the gap closing can be clearly seen in numerical calculations, as long as those peaksare well separated in the momentum space. (See Appendix D for more details.) With the positions of the valleysdetermined, the valley CN on one side of the GC is not necessarily quantized to integers since the integral of Berrycurvature is not over a closed manifold. However, the change of valley CN across the GC is always integer-valued,2since it is equal to the CN of the Hamiltonian given by patching the two low-energy effective models on the twosides of the GC at large momenta, which lives on a closed manifold. One physical consequence of the quantizedchange of valley CN is the gapless domain-wall mode (46), which can be experimentally tested with transport oroptical measurements (70). We verify the quantized change of valley CN and demonstrate the corrsponding gaplessdomain-wall mode with a tight-binding model in Appendix D.The above argument relies on the constraint that the valleys are well separated in 1BZ, preventing the two statesfrom being adiabatically connected. Without the contraint of well-defined valleys, the valleys are allowed to bemerged, and two phases with different valley CNs might be adiabatically connected. Therefore, we refer to thetopology characterized by valley CN as locally stable (21), though globally unstable. Nevertheless, we restrict allvalleys to be well-defined in our discussion and refer to the corresponding gap closing case as a TQPT.
V. ACKNOWLEDGEMENT
We are thankful for the helpful discussion with B. Andrei Bernevig, Xi Dai, F. Duncan M. Haldane, Shao-Kai Jian,Biao Lian, Xin Liu, Laurens W. Molenkamp, Zhiqiang Mao, Xiao-Qi Sun, David Vanderbilt, Jing Wang, BinghaiYan, Junyi Zhang, and Michael Zaletel. We acknowledge the support of the Office of Naval Research (Grant No.N00014-18-1-2793), the U.S. Department of Energy (Grant No. DESC0019064) and Kaufman New Initiative researchgrant KA2018-98553 of the Pittsburgh Foundation.
Appendix A: Derivation of the PET
In this section, we derive Eq. (3) in the main text via linear response theory from Eq. (2) in the main text, whichis equivalent to the derivation in Ref. (15). The derivation is done with the natural unit c = (cid:126) = 1 and the metric( − , + , +).To apply the linear response theory, we start from an action S that includes the electronic effective model and theleading order effect of the infinitesimal strain. Since the current is present in Eq. (2) , we should include the U (1)gauge field that accounts for the electromagnetic field. With the U (1) gauge field, the action reads S = (cid:90) d k (2 π ) ψ † k G − ( k ) ψ k + (cid:90) d k (2 π ) (cid:90) d q (2 π ) (cid:20) ψ † k + q/ ∂G − ( k ) ∂k µ ψ k − q/ eA µ ( q ) − ψ † k + q/ M ij ψ k − q/ u ij ( q ) (cid:21) , (A1)where k µ = ( ω, k ) µ , A µ and u ij and ψ follow the same Fourier transformation rule, G ( k ) = [ ω − h ( k )(1 − i (cid:15) )] − isthe time-ordered Green function without the electron-strain coupling, the chemical potential is chosen to be the zeroenergy, and M ij is the matrix coupled to the strain tensor u ij . To the leading order, the linear response is given bythe following effective action S eff = (cid:90) d x e∂ ν A µ u ij f ij,µν , (A2)where f ij,µν = − (cid:90) d k (2 π ) { Tr[ G ∂G − ∂k µ G ∂G − ∂k ν G M ij ] − ( µ ↔ ν ) } , (A3)and the absence of the Chern-Simons term AdA is due to the T symmetry.With Eq. (A2) and Eq. (2) , we can use the condition that u ij is uniform to derive the expression of the PET,resulting in γ ijk = − ef jk,i . (A4)To further derive Eq. (3) , we define h ( k , u ij ) = h ( k ) + u ij M ij and G ( k, u ij ) = [ ω − h ( k , u ij )(1 − i (cid:15) )] − as theHamiltonian and Green function with the electron-strain coupling, respectively. Using ∂ k µ G − = ∂ k µ G − and ∂ u ij G − = − M ij , we can revise Eq. (A4) to γ ijk = e (cid:90) d k (2 π ) { Tr[
G ∂G − ∂k i G ∂G − ∂ω G ∂G − ∂u jk ] − ( k i ↔ ω ) }| u ij → . (A5)3Define X µ = ( ω, k i , u jk ) and then the above equation can be further transformed to γ ijk = − e (cid:90) d kdω (2 π ) (cid:15) µνρ Tr[
G ∂G − ∂X µ G ∂G − ∂X ν G ∂G − ∂X ρ ] (cid:12)(cid:12)(cid:12)(cid:12) u ij → , (A6)where (cid:15) µνρ is the Levi-Civita symbol. Integrating out ω in the above equation with the Wick rotation gives Eq. (3) .Although the derivation here is done for (cid:126) = c = 1, all the expressions of γ ijk and the resultant Eq. (3) stay the sameafter converting to the SI unit as they carry the right unit for the PET in 2+1D.Finally, we would like to discuss the effect of the identity term of h in Eq. (A2) when h is a two band model.In general, the Hamiltonian can always be split into the identity part and the traceless part as h ( k ) = m ( k ) + h traceless0 ( k ). The eigenvalues of h ( k ) then read m ( k ) ± ε ( k ), where ± ε ( k ) are two eigenvalues of h traceless0 ( k ) with ε ( k ) > E = 0) is chosen to lie insidethe gap, we have ε ( k ) > | m ( k ) | ≥
0. Since the poles of G are at ω = [ m ( k ) ± ε ( k )](1 − i (cid:15) ), integrating ω along( −∞ , ∞ ) in f ij,µν of Eq. (A2) gives the same result as integrating ω along ( −∞ + m ( k )(1 − i (cid:15) ) , ∞ + m ( k )(1 − i (cid:15) ))owing to the absence of poles in between the two paths. As a result, we can directly neglect the identity term of atwo-band insulating h in f ij,µν of Eq. (A2). Appendix B: Details on PET for Each PG
The discussion on the electronic effective model and FTP of the gap closing between two non-degenerate stateshas some overlap with Ref. (45). However, the topological property and PET jump of the gap closing between twoinsulating states have not been discussed in Ref. (45).
1. PG p In the main text, the effective Hamiltonian for scenario (ii) of p h ± ( q , u ) = E ( ± q ) σ + ( v x q x + v q y ) σ x + v y q y σ y ± mσ z + ξ ,ij σ u ij ± ξ a (cid:48) ,ij σ a (cid:48) u ij . (B1)Here we only perform the unitary transformation on the bases of the Hamiltonian and do not rotate the momentumor the coordinate system. Correspondingly, the PET jump across the direct TQPT at m = 0 can be derived as∆ γ xij = − e ∆ N + π ξ y,ij v y ∆ γ yij = e ∆ N + π (cid:18) ξ x,ij v x − v v x ξ y,ij v y (cid:19) . (B2)Eq. (B1)-(B2) resemble the conclusion for p p
1. In the first scenario, all TRIM have no essentialdifferences and the gap closing always happens between two Kramers pairs unless more parameters are finely tuned.Therefore, there is no need to further classify this scenario into finer cases, and the codimension for the gap closingis 5, indicating that this scenario cannot be direct TQPT (26). According to the main text, no finer classification isneeded for the second scenario either, the codimension of the gap closing scenario is 1, and it is indeed a direct TQPTthat changes the Z index and leads to the PET jump.
2. PGs p m , c m and p g In this part, we study three PGs, p m , c m
1, and p g
1, all of which are generated by a mirror-related symmetry U and the lattice translation. U is a mirror operation for p m /c m p g
1. The difference4 𝑚 > 0 𝑚 = 0 𝑚 < 0 𝑝1𝑔1 𝐸 Y/M Y/M Y/M
Fig 4:
The figure shows the gap closing at Y or M in scenario (i) for p g
1. The lines indicate the band dispersionalong k y , and those in the same (different) colors have the same (opposite) g x eigenvalues. m labels the gap at Y or M , and when the gap at Y or M is open ( m (cid:54) = 0), the system is still in a gapless phase.between p m c m x , labelled as m x or g x , respectively. The glide operation is thus denoted as g x = { m x | } , where “0 ” representsthe translation by half the primitive lattice vector along y . The U symmetry in these three PGs requires γ xxx = γ xyy = γ yxy = γ yyx = 0, whereas the PET components γ xxy , γ xyx , γ yxx , γ yyy are allowed to be nonzero. For thesymmetry analysis here, the PET behaves the same under the glide and mirror operations since u ij is considered inthe continuum limit.In order to classify the gap closing scenarios, we define the group G for a gap closing momentum k such that G contains all symmetry operations that leave k invariant. Since G can include the TR-related operation, it canbe larger than the little group of k . Based on G , we obtain in total 4 gap closing scenarios for these three PGs:(i) the gap closing at TRIM ( G contains T ), (ii) G contains U but not T , (iii) G contains UT but not T , (iv) G contains no symmetries other than the lattice translation, which we refer to as the trivial G . As summarized in Tab. 1in the main text, the TQPT exists in scenario (iii) and (iv), which can lead to the jump of symmetry-allowed PETcomponents. a. Scenario (i): TRIM In scenario (i), the gap closing requires 3 (5) fine-tuning parameters (FTPs) for p m c m m x is (is not) inthe G . (See Appendix. C 2.) For p g
1, the TRIM (Γ, X , Y and M ) are split into two classes according to the valueof g x : Γ , X with g x = − Y, M with g x = 1. The gap closing at Γ , X needs 3 FTPs since g x behaves the sameas m x , while the gap closing at Y, M needs only 1 FTP if it happens between two Kramers pairs with opposite g x eigenvalues. However, such gap closing at Y, M is in between two g x -protected gapless phases with codimension 0,where the bands with opposite g x eigenvalues cross with each other at momenta other than Y, M as shown in Fig. 4.Therefore, there is no direct TQPT between two gapped phases in scenario (i). b. Scenario (ii):
U ∈ G but T / ∈ G The same situation occurs for scenario (ii). In scenario (ii), the gap closes at two different momenta ± k that areinvariant under the U operation, meaning that the bases at ± k can have definite U eigenvalues. The gap closingbetween the two bases with the same U eigenvalues requires 2 FTPs, as discussed in Appendix. C 2. When the gapcloses between two bands with opposite U eigenvalues, the system always enters a stable U -protected gapless phasewith 0 codimension. (This case is not the same as the scenario (i) since only one side is guaranteed to be gapless.)Thus, the gap closing cases cannot be direct TQPTs.5 c. Scenario (iii): UT ∈ G but T / ∈ G In scenario (iii), the gap closing occurs at two different momenta ± k that are invariant under UT , as shown bythe orange dashed lines in Fig. 1 (b) and (c) in the main text.For p m c m U = m x ), ( m x T ) = 1 suggests that we can have m x T ˙= K at k by choosing the appropriatebases and the band touching point at k should typically occur between two non-degenerate bands. We further take T ˙=i σ y K by choosing the appropriate bases at − k , and thus the two-band effective models h ± ( q , u ) at ± k can begiven by Eq. (B1) with extra constraints v = ξ a ,xy = ξ a ,yx = ξ y,xx = ξ y,yy = 0 (B3)for a = 0 , x, z . As a result, only 1 FTP m is needed for the gap closing ( m = 0), and only one single Dirac cone existsin half 1BZ at the transition, leading to the change of the Z index. Based on Eq. (B2), the jump of symmetry-allowedPET components across this TQPT can be derived as∆ γ xxy = ∆ γ xyx = − e ∆ N + π ξ y,xy v y ∆ γ yxx = e ∆ N + π ξ x,xx v x ∆ γ yyy = e ∆ N + π ξ x,yy v x . (B4)For p g U = g x , since ( g x T ) = 1 at ( k x ,
0) and ( g x T ) = − k x , ± π ), we have two different gap closingcases. When the gap closes at ( ± k ,x , g x T is the same as m x T , e.g. ( g x T ) =( m x T ) = 1, and thus the effective Hamiltonian can be chosen to be the same as that for p m c m
1, leading to1 FTP, Z index change, and the same form of PET jump. On the contrast, due to ( g x T ) = − ± k ,x , ± π ), thegap closing needs 4 FTPs and thus no TQPT can occur in this case. (See Appendix. C 2.) d. Scenario (iv): trivial G In scenario (iv), the gap should close simultaneously at four momenta k , k = − k , k = U k , and k = −U k , asdepicted in Fig. 1 (d) and (e) in the main text. The gap closing at k can be described by the Hamiltonian h + ( q , u )in Eq. (B1), and the Hamiltonian at k , k , and k can be given by T h + ( − q , u ) T † , U h + ( U − q , U − u ( U − ) T ) U † , and UT h + ( −U − q , U − u ( U − ) T )( UT ) † , respectively. Therefore, the gap closing can be achieved by tuning 1 FTP, i.e. m in h + ( q , u ), in this scenario.There is no change of Z index for this scenario, since two Dirac cones exist in half 1BZ when the gap closes and theCN of contracted half 1BZ can only change by an even number. Nevertheless, scenario (iv) can still be “topological” inthe context of valley Chern number (VCN) as elaborated in the following. Due to the Dirac Hamiltonian form shownin Eq. (B1), the Berry curvature is peaked at each valley k , , , for a small m and can be captured by the electronicpart of the corresponding effective Hamiltonian. Then, we can integrate the Berry curvature given by the effectivemodel and get the VCN (20, 25) for each valley as N k i = − η i sgn( v x v y )sgn( m ) / i = 0 , , ,
3. The values of η i at different valleys are related by the TR and U symmetries, both of which flip the sign of the Berry curvature. Thus,we have η = η = 1 and η = η = −
1. It should be pointed out that the Berry curvature integral is not over theentire 1BZ and the VCN at each valley thus does not need to be an integer. Nevertheless, the change of VCN acrossthe gap closing is defined on a closed manifold and must be an integer number, given by ∆ N k i = − η i sgn( v x v y ) asvarying m from 0 − to 0 + . For the convenience of further discussion, we can define the VCN of the whole system (25)as N val = (cid:80) i η i N k i = − v x v y )sgn( m ), and the change of the VCN becomes ∆ N val = − v x v y ) = 4∆ N + withthe factor 4 for the four valleys. Therefore, if we restrict all the valleys to be far apart in the momentum space, thechange of the VCN is a well-defined topological invariant and this gap closing scenario is a TQPT.In principle, tuning parameters may merge different valleys at some high symmetry momentum, e.g. the valleysat k and U k merged at the mirror or glide line. Therefore, without the constraint of well-defined valleys, twophases with different VCNs can share the same band topology and thus can be adiabatically connected. It means thetopology characterized by VCN is “locally stable” (21), though globally unstable. Nevertheless, we restrict all valleysto be well-defined in our discussion and refer to the gap closing scenario as a TQPT.Next we study the change of the PET components at this TQPT, which can be split into two parts: ∆ γ (0) originatingfrom ± k and ∆ γ (1) given by ±U k . ∆ γ (0) equals to Eq. (B2) since the effective models at ± k are the same as6 (b)(a) 𝑣 /(eV Å) Δ𝛾 𝑦𝑦𝑦 e Å −1 𝑚/eV 𝛾 𝑦𝑦𝑦 e Å −1 Δ𝛾 𝑥𝑥𝑥 e Å −1 𝑣 /(eV Å) 𝑚/eV 𝛾 𝑥𝑥𝑥 𝛾 𝑦𝑦𝑦 ( e Å −1 ) 𝑣 /(eV Å) Fig 5:
The upper panels of ( a ) and ( b ) numerically show the PET jump induced by the gap closing at Γ point asthe function of v , and v , respectively. The lower panels of (a) and (b) plot the PET component as a function of m for ( v , v ) = (0 . , . v = 0 . γ (1) is related to ∆ γ (0) as ∆ γ (1) ijk = ( U ) ii (cid:48) ( U ) jj (cid:48) ( U ) kk (cid:48) γ (0) i (cid:48) j (cid:48) k (cid:48) . As aresult, we obtain the non-zero jump of symmetry-allowed PET components ∆ γ ijk = ∆ γ (0) ijk + ∆ γ (1) ijk as∆ γ xxy = ∆ γ xyx = − e ∆ N val πv y ξ yxy ∆ γ yxx = − e ∆ N val ( − v y ξ xxx + v ξ yxx )2 πv x v y ∆ γ yyy = − e ∆ N val ( − v y ξ xyy + v ξ yyy )2 πv x v y . (B5)
3. PG p PG p C and the lattice translation. Owing to C , the PET only has two independentcomponents γ xxx and γ yyy as γ yxy = γ yyx = γ xyy = − γ xxx γ xxy = γ xyx = γ yxx = − γ yyy . (B6)Again, we classify the gap closing for p G , resulting in three different scenarios: (i) G contains T , (ii) G contains C but not T , and (iii) G is trivial. Here we do not have a scenario for G containing C T but no T ,since ( C T ) is equivalent to T . As summarized in Tab. 1 in the main text and elaborated in the following, in any ofthe above scenarios, there are gap closing cases between gapped states that need only 1 FTP, change the Z index,and lead to the discontinuous change of symmetry-allowed PET components. a. Scenario (i):TRIM There are 4 TRIM in scenario (i), namely three M points related by C and one Γ point, as labeled in Fig. 1 (f) inthe main text. G of each individual M point only contains T and the lattice translation, and thus the gap closing at M needs 5 FTPs, same as the gap closing at TRIM for p G also contains C with C = −
1. Due to[ C , T ] = 0, the Kramers pairs can be classified into two types according to the C eigenvalues: one with (e − i π/ , e i π/ )and the other with ( − , − − i π/ , e i π/ ) type and 5 for ( − , −
1) type, as discussed in Appendix. C 3.The gap closing with 1 FTP happens between the TR pairs of different types, for which the minimal four-bandeffective Hamiltonian in the bases (e − i π/ , e i π/ , − , −
1) reads h p ( k , u ) = h p , ( k ) + h p , ( u ) , (B7)7where h p , is the electron part h p , ( k ) = E τ σ + mτ z σ + ( v k x + v k y )( τ + τ z σ x + ( v k y − v k x )( τ + τ z σ y + ( v k x + v k y ) τ x σ z + ( − v k y + v k x ) τ y σ + ( v k x + v k y ) τ x σ x + ( − v k y + v k x ) τ x σ y , (B8)and h p , describes the electron-strain coupling h p , ( u ) = ( u xx + u yy )( ξ τ σ + ξ τ z σ )+ ( − u xx + u yy )( ξ τ y σ z + ξ τ y σ x − ξ τ x σ + ξ τ y σ y )+ ( u xy + u yx )( ξ τ y σ z + ξ τ y σ x + ξ τ x σ − ξ τ y σ y ) . (B9) τ ’s and σ ’s are Pauli matrices that label two different Kramers pairs and two components of each Kramers pair,respectively, m is the gap closing tuning parameter, and the bases are chosen such that T ˙= − i τ σ y K .This gap closing is certainly a TQPT since it changes the numbers of IRs of the occupied bands, meaning that thetwo gapped states separated by this gap closing cannot be adiabatically connected. When v = v = 0, we can definean effective inversion symmetry (cid:101) P = τ z σ for the electron part of Eq. (B7), (cid:101) P h p , ( − k ) (cid:101) P † = h p , ( k ), and thus thegap closing of h p , ( k ) with v = v = 0 changes the Z index according to the Fu-Kane criteria (71) since the parityof the occupied band changes. The existence of non-zero v , v terms that break (cid:101) P cannot influence the Z topologychange, since (i) the Z topology does not rely on the effective inversion symmetry, and (ii) additional gap closingaway from Γ is forbidden at m = 0 as long as the v , terms are restored adiabatically. Therefore, within a certainrange of v , , the codimension-1 gap closing at m = 0 is a direct TQPT that changes the Z index.The remaining question is if the codimension-1 gap closing at m = 0 is always a Z transition. To answer thisquestion, note that we can always assume the transition at m = 0 is Z for a parameter region S of v i ’s in h p , andnon- Z for the other parameter region S of v i ’s. Since the same form of the Hamiltonian can not correspond to Z and non- Z transitions simultaneously, the intersection of S and S is empty. Now we suppose both S and S arecodimension-0 subspaces of the v i parameter space (not the whole parameter space since only v i ’s are included while m is excluded). Then, the boundary of S , labeled as ∂S , is a codimension-1 subspace of v i parameter space, andthe special transition at ( m = 0 , v i ∈ ∂S ) is a codimension-2 transition, as shown in Fig. 6.Patching the Hamiltonian with m = ± (cid:15) with (cid:15) positive and infinitesimal gives a Hamiltonian that lives on a closedmanifold. This Hamiltonian has Z trivial and nontrivial ground states when v i ’s are in S and S , respectively, asshown in Fig. 6. As v i ’s change from S to S passing through ∂S , the patched Hamiltonian must experience a gapclosing at generic k points that changes Z (Fig. 6), since there is always an energy gap at Γ for m (cid:54) = 0. As discussedwith more detail in the following (see scenario (iii)), the gap closing at generic k points surely changes the Z index,is codimension-1, and simultaneously happens at six momenta. The gap closing can only happen either for m = (cid:15) part or for m = − (cid:15) part of the patched Hamiltonian but not both, since if the gap closes twice, the Z index would bechanged back. It means, the codimension-1 hypersurface for the gap closing at generic k (red line in Fig. 6) touchesthe codimension-1 hypersurface for gap closing at Γ ( m = 0 line in Fig. 6) just from one side of m but not passingthrough. As mentioned above, the touching part at ( m = 0 , v i ∈ ∂S ) is a codimension-2 transition, owing to theassumption that both S and S are codimension-0 subspaces of v i parameter space.At the touching, we must have the six generic gap closing points merging at Γ. Otherwise, we should expect the redline in Fig. 6 to pass through the m = 0 line instead of stopping, since the gap closing process is local in the momentumspace and different gap closing cases cannot influence each other if they happen at the different momenta. However,the merging process cannot be codimension-2 since moving a generic gap closing point to a specific momentum whilekeeping the gap closed requires at least 3 FTPs (two to move the momentum and one to close gap). Therefore, S and S cannot be both codimension-0 subspaces of v i parameter space, and the codimension-1 gap closing at Γ canonly be Z or non- Z but not both. Since we already show that the Z transition at Γ can be codimension-1, thecodimension-1 gap closing at Γ should always change the Z index.We next study the non-zero PET components, starting from the v = v = 0 case. If we further set ξ = ξ = ξ = ξ = 0, the electron-strain coupling h p , also has the effective inversion (cid:101) P , leading to the vanishing PET. It means8 𝑚 {𝑣 𝑖 }𝑆 𝑆 𝑍 trivial 𝑍 trivial 𝑍 nontrivial A C B Fig 6:
This figure shows the typical phase diagram around ( m = 0 , v i ∈ ∂S ) (the black dot), within theassumption that both S and S are codimension-0 subspaces of the v i parameter space. The green line is( m = 0 , v i ∈ S ) which does not change Z index, the blue line is ( m = 0 , v i ∈ S ) that changes Z index, and thesystem closes the gap at six generic k points on the red line. Without loss of generality, we choose the red line totouch the m = 0 line from the positive m side. A,B, and C label three different Hamiltonians given by patching thetwo effective models with m = ± (cid:15) at different values of v i . A is Z trivial, C is Z non-trivial, and B closes the gapat the six generic k points for in m = (cid:15) part.that ξ and ξ cannot contribute to the PET for v = v = 0. Indeed, the direct derivation gives the PET jump∆ γ xxx = − eπ (cid:80) b =3 v b ξ b (cid:80) b =3 v b ∆ γ yyy = eπ v ξ − v ξ + v ξ − v ξ (cid:80) b =3 v b . (B10)For non-zero v and v , the PET components can be calculated numerically for v = v = v = v = 1eV˚A and ξ = ξ = ξ = 2 ξ = ξ = 2 ξ = 1eV, showing the jump across the TQPT in Fig. 5(a). b. Scenario (ii): C ∈ G and T / ∈ G The gap closing momenta in scenario (ii) are K and K (cid:48) in Fig. 1 (g) in the main text. Since these two momenta arerelated by T , we only need to derive the effective model at one momentum, say K , and the other one can be obtainedusing T . At K , the C symmetry has three possible eigenvalues − , e ± i π/ due to C = −
1. If the gap closing isbetween two states with the same C eigenvalues, it cannot be TQPT since the fixed gap closing momentum leads to3 FTPs for the gap closing. (See Appendix. C 3.)There are three cases for two states with different C eigenvalues: (e − i π/ , e i π/ ), (e i π/ , − − , e − i π/ ). Theeffective models in the three cases are equivalent since the representations of C in these cases can be related to eachother by multiplying a phase factor e ± i2 π/ . Therefore, we focus on the first case, of which the effective model at K (after an appropriate unitary transformation) is given by h + in Eq. (B1) with v x = v y ≡ v, v = 0 , ξ a (cid:48)(cid:48) ,xy = ξ a (cid:48)(cid:48) ,yx = 0 ,ξ a (cid:48)(cid:48) ,xx = ξ a (cid:48)(cid:48) ,yy , ξ x,xx = − ξ x,yy = − ξ y,xy = − ξ y,yx ,ξ y,xx = − ξ y,yy = ξ x,xy = ξ x,yx , (B11)where a (cid:48)(cid:48) = 0 or z . Similarly, by choosing the appropriate bases at K (cid:48) such that T ˙= iσ y K , the effective model at K (cid:48) is given h − in Eq. (B1) with the parameter relation listed above. As a result, the gap closing between states with9different C eigenvalues needs 1 FTP and changes the Z index since half 1BZ contains one Dirac cone ( K or K (cid:48) ).Based on Eq. (B2), the jump of independent PET components across this TQPT (varying m from 0 − to 0 + ) has thenon-zero form ∆ γ xxx = − e ∆ N + ξ y,xx πv , ∆ γ yyy = e ∆ N + ξ x,yy πv , (B12)where ∆ N + = − sgn( v ) = − c. Scenario (iii): trivial G In scenario (iii), there are six gap closing momenta, labeled as ± k , ± C k and ± C k , as shown by red crosses inFig. 1 (h) in the main text. The effective Hamiltonian at ± k are exactly the same as Eq. (B1) since the two momentaare related by T and no more symmetries are involved. Therefore, the gap closing scenario needs 1 FTP, and thecontribution to the PET jump from the gap closing at ± k is the same as Eq. (B2), noted as ∆ γ (0) ijk . The effective modelsat ± C k and ± C k can be obtained from those at ± k by C and C operations, respectively, whose electronicparts are also in the Dirac Hamitlonian form. The contracted half 1BZ then contains three Dirac cones at the gapclosing and its CN must change by an odd number, indicating the change of Z index. Furthermore, the contributionsto the jump of PET components from the gap closing at ± C k and ± C k are ∆ γ (1) ijk = ( C ) ii (cid:48) ( C ) jj (cid:48) ( C ) kk (cid:48) ∆ γ (0) i (cid:48) j (cid:48) k (cid:48) and ∆ γ (2) ijk = ( C ) ii (cid:48) ( C ) jj (cid:48) ( C ) kk (cid:48) ∆ γ (0) i (cid:48) j (cid:48) k (cid:48) , respectively, owing to the symmetry. As a result, the jump of independentPET components is given by ∆ γ ijk = ∆ γ (0) ijk + ∆ γ (1) ijk + ∆ γ (2) ijk , which has the nonzero form∆ γ xxx = − e N + π v y ξ x,xy − v ξ y,xy + v x ( ξ y,xx − ξ y,yy ) v x v y ∆ γ yyy = e N + π (cid:20) v y ( − ξ x,xx + ξ x,yy ) + 2 v x ξ y,xy v x v y + v ( ξ y,xx − ξ y,yy ) v x v y (cid:21) . (B13)
4. PG p m and PG p m Both PGs p m and p m C , and a mirror symmetrywhich we choose to be m x without loss of generality. The difference between the two PGs lies on the direction of themirror line relative to the primitive lattice vector: the mirror line is parallel or perpendicular to one primitive latticevector for p m or p m
1, respectively. C and m x span the point group C v , which leads to γ xxx = γ xyy = γ yxy = γ yyx = 0 γ xyx = γ xxy = γ yxx = − γ yyy (B14)for the PET, and thus γ yyy serves as the only independent symmetry-allowed PET component. We classify the gapclosing scenarios into 4 types according to G : (i) G contains T , (ii) G contains at least one of the three mirrorsymmetry operations in C v (again labeled as U = m x , C m x , or C m x ) but no T , (iii) G contains the UT but no T , and (iv) G is trivial. As summarized in Tab. 1 in the main text, all gap closing cases between gapped states with1 FTP change either Z index or the VCN, and lead to the jump of symmetry-allowed PET components. a. Scenario (i): TRIM Similar as Sec. B 3 a for PG p
3, there are four inequivalent TRIM: the Γ point and three M points. Although G atthe M point now contains U , the gap closing still requires 3 FTPs same as the corresponding case in Sec. B 2 a, whichcannot be a TQPT.When the gap closes at Γ point (Fig. 1 (i-j) in the main text), the generators of G besides the lattice translationare C , m x and T , and there are still two types of Kramers pairs characterized by the C eigenvalues as those inSec. B 3 a. Owing to the extra mirror symmetry here, the number of FTPs for the gap closing between the same typeof Kramers pairs becomes 2 for (e − i π/ , e i π/ ) type and 3 for ( − , −
1) type as discussed in Appendix. C 4. Therefore,0we still only need to consider the gap closing between different types of Kramers pairs. For the convenience of thelater material discussion, we choose the bases as (e − i π/ , − , e i π/ , − T ˙= − i σ y τ K and m x ˙= − i σ x τ . In this case, the effective Hamiltonian can bederived by imposing the m x on Eq. (B7), leading to v = v = v = ξ = ξ = 0 . (B15)The form of the Hamiltonian then reads h p m ( k , u ij ) = E + ξ u + m v k + i v k − − i v k + v k − − m − i v k + − i v k + i v k − m − v k − i v k − − v k + − m + ξ u ξ u − ξ u − ξ u + − ξ u − i ξ u −
00 i ξ u + ξ u ξ u + − i ξ u + ξ u − − ξ u , (B16)where u = u xx + u yy , u ± = u xx − u yy ± i( u xy + u yx ).The above Hamiltonian shows that the gap closing at Γ needs only 1 FTP, which is m . As discussed in Appendix. C 4,this gap closing cannot drive a gapped phase into a mirror-protected gapless phase, and therefore can separate twogapped states. Similar to the discussion in Sec. B 3 a, the gap closing changes the Z index when tuning m from 0 − to 0 + , indicating a TQPT. When v = 0, an analytical expression for the jump of independent PET component canbe obtained from Eq. (B10) and Eq. (B15), which reads∆ γ yyy = eπ − v ξ + v ξ v + v . (B17)With parameter values v = v = 1eV˚A and ξ = ξ = 2 ξ = ξ = 1eV, the numerical results (Fig. 5(b)) for non-zero v still show a PET jump across TQPT. b. Scenario (ii): U ∈ G and T / ∈ G Scenario (ii) can be further divided into two classes depending on whether G contains C . When G does notcontain C , the gap closing either requries more than 1 FTP or drives the system into a mirror-protected gaplessphase with 0 codimension, similar to Sec. B 2 b.Only when the gap closes at K, K (cid:48) for p m , G contains C . In this case, G contains the group C v , which hasone 2D irreducible representation (IR) and two different 1D IRs when acting on the states. The gap closing betweenthe states furnishing the same IR requires 3 FTPs, similar to the case for two states with the same C eigenvalue inSec. B 3 b. If the gap closes between the doubly degenerate states furnishing the 2D IR and a state furnishing a 1DIR, the system with a fixed carrier density cannot be insulating on both sides of the gap closing because the numberof occupied bands is changed. If the gap closes between two states that furnish different 1D IRs, the mirror-protectedgapless phase must exist on one side of the gap closing as the two states must have opposite mirror eigenvalues.Therefore, there is no direct TQPT between the insulating phases in scenario (ii). c. Scenario (iii): UT ∈ G and T / ∈ G In scenario (iii), the gap closing cases are again divided into two different classes depending on whether G has C . We first discuss the class without C , which happens for the gap closing at UT invariant momenta except K, K (cid:48) for p m
1. As shown in Fig. 1 (k-m) in the main text, the total number of inequivalent gap closing momenta is six,including ± k , ± C k , and ± C k . Without loss of generality, we choose k such that − m x k is equivalent to k .1Then, the effective models at ± k are the same as the corresponding models in Sec. B 2 c, i.e. Eq. (B1) with theparameter relation Eq. (B3), indicating 1 FTP for the gap closing. Since the effective models at ± C k and ± C k are related to those at ± k by C and C operations similar to Sec. B 3 c, the jump of PET components can be derivedby substituting Eq. (B3) into Eq. (B13), resulting in∆ γ yyy = e N + π v y ( − ξ x,xx + ξ x,yy ) + 2 v x ξ y,xy v x v y . (B18)Moreover, since three Dirac cones exist in half 1BZ when the gap closes, the Z index changes at the gap closing,making it a TQPT.The class that G includes C can only happen when the gap closes at K and K (cid:48) for PG p m
1, as shown in Fig. 1(m) in the main text. We can choose C and m x T as the generators of G besides the lattice translation. Similar toSec. B 3 b, we first study K and derive the model at K (cid:48) by choosing the right bases such that T ˙=i σ y K . The states at K can be labeled by C eigenvalues, − ± i π/ given by C = −
1. Since ( m x T ) = 1 and C m x T = m x T C − , thegap closing typically happens between two non-degenerate states, labeled by the C eigenvalues as ( λ , λ ), and we canalways choose m x T ˙= K . The λ = λ case cannot correspond to TQPT since 2 FTPs are needed for the gap closing asdiscussed in Appendix. C 4, while the λ (cid:54) = λ case requires only one FTP for the gap closing similar to Sec. B 3 b. Sincethe matrix representations of C and m x T are equivalent for the three choices ( λ , λ ) = (e − i π/ , e i π/ ) , (e i π/ , − , and ( − , e − i π/ ), they have the same effective models and we only consider the first choice. With all the aboveconventions and simplifications, the effective models at K and K (cid:48) can be given by those for Sec. B 3 b with an extraconstraint ξ y,yy = 0 brought by m x T . As a result, the Z index does change when the gap closes, and the jump ofPET components can be derived from Eq. (B12) with the above extra constraint, which reads∆ γ yyy = − e ∆ N + ξ x,xx πv . (B19) d. Scenario (iv): trivial G In scenario (iv), the gap closes simultaneously at twelve inequivalent momenta, namely ± k , ± m x k , ± C k , ± C m x k , ± C k and ± C m x k in Fig. 1 (n-o) in the main text. The effective model around k can be chosen as h + in Eq. (B1), and the models around other gap closing momenta can be further obtained by the symmetry. Althoughthis gap closing scenario only needs 1 FTP, it cannot induce any change of Z index since there is an even number(six) of Dirac cones in contracted half 1BZ. However, the gap closing can change the VCN when the twelve valleysare well defined according to Appendix. B 2 d, e.g. N k can change by ±
1, and thus is a TQPT in the sense of thelocally stable topology.We split the change of PET components for this scenario into 3 parts: γ (0) from ± k and ± m x k , γ (1) from ± C k and ± C m x k , and γ (2) from ± C k and ± C m x k . Since the contribution to γ (0) contains twoKramers pairs that are related by m x , same as Sec. B 2 d, γ (0) equals to Eq. (B5). C symmetry then gives∆ γ (1) ijk = ( C ) ii (cid:48) ( C ) jj (cid:48) ( C ) kk (cid:48) ∆ γ (0) i (cid:48) j (cid:48) k (cid:48) and ∆ γ (2) ijk = ( C ) ii (cid:48) ( C ) jj (cid:48) ( C ) kk (cid:48) ∆ γ (0) i (cid:48) j (cid:48) k (cid:48) , similar to Sec. B 3 c. As the re-sult, the total change of PET can be obtained from ∆ γ = ∆ γ (0) + ∆ γ (1) + ∆ γ (2) , which is propotional to the changeof the VCN of the system ∆ γ yyy = e ∆ N val π (cid:20) v y ( − ξ x,xx + ξ x,yy ) + 2 v x ξ y,xy v x v y + v ( ξ y,xx − ξ y,yy ) v x v y (cid:21) (B20)with ∆ N val = 12∆ N + .
5. 10 PGs with 2D Inversion or C The PET jump cannot exist in 10 PGs that contain C or inversion, including p p mm , p mg , p gg , c mm , p p mm , p gm , p
6, and p mm . This conclusion can be drawn from the symmetry analysis of PET. Since both C andinversion transform ( x, y ) to ( − x, − y ), γ ijk = − γ ijk is required for those 10 PGs, leading to the vanishing PET. Earlystudy(21, 44) also shows that a stable gapless phase can exist in between the QSH insulator and the NI when C exists. In this gapless regime, 2D gapless Dirac fermions are locally stable and can only be created or annihilated inpairs.2 Appendix C: Numbers of FTPs and Effective Models for the Gap Closing
The discussion on the gap closing between two non-degenerate states has some overlap with Ref. (45).
1. PG p This part has been studied in Ref. (26). a. Not TRIM
When the gap closes at k that is not a TRIM, the two-band model near the gap closing to the leading order of q = k − k in general takes the form h ( q ) = E ( q ) σ + ( q x C x + q y C y + M ) · σ , (C1)where C i = ( C ix , C iy , C iz ), M = ( M x , M y , M z ), σ = ( σ x , σ y , σ z ), and the two bases of the above model account forthe doubly degenerate band touching when the gap closes. Eq. (C1) determines the codimension for the gap closingscenario since the gap at − k is related to that of Eq. (C1) by the TR symmetry. The gap of Eq. (C1) closes if andonly if C x q x + C y q y + M = 0. We choose C x × C y (cid:54) = 0 since it can be satisfied without finely tuning anything (orequivalently in a parameter subspace with 0 codimension). In this case, the gap closes when M lies in the planespanned by two vectors C x and C y . Therefore, the codimension for the gap closing is 1 since only the angle betweenthe vector M and the ( C x , C y ) plane needs to be tuned.Next, we derive Eq. (5) of the main text and the electronic part of h + in Eq. (B1), while the model at − k can be derived by the TR symmetry and thus is not discussed here. Eq. (C1) always allows the q -independent SU(2)transformation, i.e. h ( q ) → U † h ( q ) U with U ∈ SU(2). Such transformation only changes the bases of the Hamiltonianbut does not change the direction of the momentum or the coordinate system. Since σ behaves as an SO(3) vectorunder U , every SU(2) transformation U of the Hamiltonian is equivalent to an SO(3) transformation R of the vectors C i and M , i.e. C i → R C i and M → R M . Thus, by choosing appropriate U matrix, we can first rotate C x to the x direction and then C y to the xy plane, resulting in R C x = v x e x , R C y = v e x + v y e y and R M = m e x + m e y + m e z .As a result, Eq. (C1) is transformed to h ( q ) = E ( q ) σ + ( v x q x + v q y + m ) σ x + ( v y q y + m ) σ y + mσ z . (C2)Here C x × C y (cid:54) = 0 gives non-zero v x and v y . With a shift of k by ( m /v x − v m / ( v x v y ) , m /v y ), the model isfurther simplified to the electronic part of h + in Eq. (B1). Finally, we define the q x + v q y and q y to be q and q ,respectively, to get Eq. (5) , which represents the most generic form of the Hamiltonian. b. TRIM In this part, we count the number of FTPs for the gap closing at the TRIM. Owing to the Kramers’ degeneracy,every band at the TRIM is doubly degenerate, and we use the name ”Kramers pair” to label the two states related bythe TR symmetry. We consider the gap closing between two Kramers pairs | , ±(cid:105) and | , ±(cid:105) , where T | i, + (cid:105) = −| i, −(cid:105) can always be chosen by the unitary transformation. As a result, the mass term for the effective model at the TRIMreads m + i∆ i∆ + ∆ m i∆ − ∆ ∆ − i∆ ∆ − i∆ − i∆ − ∆ m − i∆ + ∆ ∆ + i∆ m (C3)where the bases are ( | , + (cid:105) , | , −(cid:105) , | , + (cid:105) , | , −(cid:105) ) and all parameters are real. Since the momentum is fixed at TRIM( − k = k + G with G a reciprocal lattice vector), none of the terms in the above equation can be canceled by shiftingthe momentum. Therefore, 5 FTPs are needed for the gap closing.3 p m , c m , and p g a. Scenario (i): TRIM If G does not contain U , which can occur on the edge of 1BZ for c m
1, the situation is the same as the TRIM inAppendix. C 1, which requires 5 FTPs. When G contains U , we should discuss the U = m x case ( p m c m U = g x case ( p g p m c m
1, since m x = −
1, two states of one Kramers pair have opposite mirror eigenvalues ± i, labeledby m x | i, ±(cid:105) = ± i | i, ±(cid:105) . On the bases ( | , + (cid:105) , | , −(cid:105) , | , + (cid:105) , | , −(cid:105) ), the effective model around the gap closing betweentwo Kramers pairs can be given by Eq. (C3) with ∆ = ∆ = 0, since the bases with different mirror eigenvaluescannot be coupled by the mass terms. As a result, 3 FTPs are needed for such gap closing scenario.For p g g x = − X and the number of FTPs for the gap closing is thus the same as the above case,which is 3. At Y and M , g x = 1 and two states of one Kramers pair have the same g x eigenvalue, 1 or −
1. In thiscase, the gap closing between two Kramers pairs with the same g x eigenvalue needs 5 FTPs, which is the same asthe TRIM scenario in Appendix. C 1. On the other hand, between two Kramers pairs with opposite g x eigenvalues,only 1 FTP needs to be tuned to close the gap at Y or M , since the off-diagonal terms (∆ , , , ) in Eq. (C3) are allforbidden. b. Scenario (ii): U ∈ G but T / ∈ G In scenario (ii), there are two gap closing momenta ± k that are related by the TR symmetry. Therefore, we onlyneed to consider one of them, say k , to derive the number of FPTs for the gap closing. At k , the states can belabeled by the eigenvalues of U . If the gap closing between two states with the same U eigenvalues, the effective modelcan be described by Eq. (C1) with | C x | = 0. The gap closes if and only if C y q y + M = 0, realizable by making twovectors M and C y parallel. Such realization needs 2 FTPs, e.g. the two components of the projection of M on theplane perpendicular to C y .When the gap closes between two states with different U eigenvalues, the effective model along the U -invariantline ( q x = 0) reads h ( q ) = E ( q y ) + ( m + Cq y + B q y ) σ z , which, by shifting the k ,y , can be simplified to h ( q ) = E ( q y ) + ( m + Bq y ) σ z . The gap for this Hamiltonian keeps closing when mB ≤
0, indicating a stable gapless phaseprotected by U with 0 codimension. c. Scenario (iii): UT ∈ G but T / ∈ G In this scenario, we here only consider the ( UT ) = − UT pair as the two degenerate states related by UT , in analog to the Kramerspair defined in Appendix. C 1. Similar as Eq. (C3), there are 5 mass terms for the gap closing between two UT pairs.However, the case here is different from the TRIM scenario in Appendix. C 1, since q x does not change under UT andthus the corresponding terms have the same form as the mass terms in Eq. (C3). One of the five mass terms can thenbe canceled by shifting k ,x , resulting in 4 FTPs for the gap closing. p a. Scenario (i):TRIM We first discuss the gap closing at Γ between two Kramers pairs of the same type. If the bases have C eigenvalues(e − i π/ , e i π/ , e − i π/ , e i π/ ), the mass term of the effective model is given by Eq. (C3) with ∆ = ∆ = 0 since thebases with different C eigenvalues cannot be coupled, resulting in 3 FTPs for the gap closing. If the bases have C eigenvalues ( − , − , − , − − i π/ , e i π/ , − , − E IR are not Hermitian. It means given two copies of ( E, +) or ( E, − ) IR, say( τ + ( σ x − i σ y ) , τ + ( σ x + i σ y )) and ( k x − i k y , k x + i k y ) furnishing ( E, − ) IR, the coefficients used for the tensor productcan be complex, e.g. c [ τ + ( σ x − i σ y )][ k x − i k y ] ∗ + c ∗ [ τ + ( σ x + i σ y )][ k x + i k y ] ∗ with complex c .4 IR Expressions A , + τ + σ , τ − σ , u + u A , − τ + σ z , τ − σ x , τ − σ y , τ − σ z E, + ( τ y σ z − i τ x σ , τ y σ z + i τ x σ ), ( τ y σ x + i τ y σ y , τ y σ x − i τ y σ y ), ( − u xx + u yy − i( u xy + u yx ) , − u xx + u yy + i( u xy + u yx )) E, − ( τ + ( σ x − i σ y ) , τ + ( σ x + i σ y )), ( τ x σ z + i τ y σ , τ x σ z − i τ y σ ), ( τ x σ x + i τ x σ y , τ x σ x − i τ x σ y ), ( k x − i k y , k x + i k y ) Table 2:
The irreducible representations (IRs) of C and TR symmetries. In A IR, the C eigenvalue of the basesis 1 and ± are parity under TR. “ E, ± ” label two 2D IRs, where the two components have the C eigenvalues(e i2 π/ , e − i2 π/ ) and transform as ± σ x K under the TR symmetry. τ ± = ( τ ± τ z ) / b. Scenario (ii): C ∈ G and T / ∈ G Here we consider the gap closing between two states with the same C eigenvalues at K or K (cid:48) . In general, the massterms at one gap closing momentum are m x σ x + m y σ y + m z σ z . Since the gap closing momentum is fixed, none of thethree mass terms can be canceled by shifting the momentum, and hence there are 3 FTPs for the gap closing. p m and p m a. Scenario (i): TRIM When the two Kramers pairs carry C eigenvalues as (e − i π/ , e i π/ , e − i π/ , e i π/ ), the effective model equals toEq. (C3) with ∆ = ∆ = 0 before considering m x , similar to the correspond case in Appendix. C 3. As m x ˙= − i σ x for each Kramers pair, the ∆ is also forbidden, resulting in 2 FTPs for the gap closing. On the other hand, if C eigenvalues are all −
1, the effective model equals to Eq. (C3) before considering m x , similar to the correspond case inAppendix. C 3, and including m x makes ∆ = ∆ = 0, leading to 3 FTPs for the gap closing.The construction of the effective model for the bases (e − i π/ , − , e i π/ , −
1) is the same as that for the (111)HgTe/CdTe quantum well, which is discussed in Appendix. E. Next we show that the gap closing at Γ in this casecannot drive a gapped system to the mirror protected gapless phase. Since the three mirror lines are related by the C symmetry, we only need to consider one of them, say k x = 0 that is invariant under m x . The eigenvalues alongthis line read E αβ = E + α k y v β (cid:114) ( m + α v k y + k y ( v + v ) (C4)with α, β take ± . E ± β bands cross at Γ and belong to the same set of connected bands. The mirror eigenvalue of the E αβ band is − α i, and then the mirror protected gapless phase happens when E ++ crosses with E −− or E + − crosseswith E − + . Both band crossings require the same condition | v k y | = (cid:88) α (cid:114) ( m + α v k y + k y ( v + v ) , (C5)since they are related by the TR symmetry. However, the above equation has no solution when m (cid:54) = 0 and v + v (cid:54) = 0.It can be seen from the inequality (cid:112) ( a + b ) + c + (cid:112) ( a − b ) + c > | b | , which holds unless c = 0 and | a | ≤ | b | .Therefore, without finely tuning more parameters to realize v + v = 0, a gapped system remains when the sign of m flips. b. Scenario (iii): UT ∈ G and T / ∈ G Here we discuss the case when the gap closes at K and K (cid:48) for PG p m C eigenvalues. Before considering m x T , the mass terms at K are m x σ x + m y σ y + m z σ z since C does not provide anyconstraints and the fixed gap closing momentum cannot be shifted to cancel any of them. Since m x T can be chosenas σ K , m y is forbideen and the remaining two mass terms serve as the 2 FTPs for the gap closing.5 𝜆 = −0.02𝜆 = 0.02(ത11) 𝑘𝐸 𝜆 VCN 𝑘 𝑥 𝑘 𝑦 𝑘 𝑥 𝑘 𝑦 𝐸 𝑘 𝑥 𝑘 𝑦 𝐸 𝑘 𝑥 𝑘 𝑦 𝐸 (a) (b) (c)(d) (e) 𝜆 = 0.02 𝜆 = 0 𝜆 = −0.02 Fig 7: (a) shows the energy dispersion of the TB model Eq. (D3) for λ = 0 . , , − .
02. The gap is zero for λ = 0.(b) shows the distribution of the Berry curvature for λ = 0 .
02. The peaks indicate the locations of valleys. (c)shows the integration of Berry curvature divided by 2 π over the k x,y > λ = − .
02 and λ = 0 .
02, respectively. (e) plots the energydispersion of domain-wall modes (modes with considerable probability near the interface in (d)). Here we choose thenumber of unit cells along (¯11) to be 70 for each part of the domain wall.
Appendix D: VCN in Tight-Binding Model
In this section, we discuss the quantization and physical meaning of the VCN change in a tight-binding (TB) modelwith p m
1. We consider a square lattice and each unit cell only contains one atom. Without loss of generality, we setthe lattice constant to 1, and choose the mirror symmetry as m y . On each atom, we include a spinful s and a spinful p y orbitals, meaning that the bases can be labeled as | R , α, s (cid:105) with R the lattice vector, α = s, p y for orbital, and s = ↑ , ↓ for spin. The bases with specific Bloch momentum can be obtained by the following Fourier transformation | k , α, s (cid:105) = 1 √ N (cid:88) k e i k · R | R , α, s (cid:105) . (D1)Then, the representations of the symmetries read m y | k , α, s (cid:105) = | m y k , α (cid:48) , s (cid:48) (cid:105) ( τ z ) α (cid:48) α ( − i σ y ) s (cid:48) s T | k , α, s (cid:105) = | − k , α, s (cid:48) (cid:105) (i σ y ) s (cid:48) s , (D2)where τ and σ are Pauli matrices for orbital and spin.With on-site terms and nearest-neighbor hopping terms, we choose the following symmetry-allowed expression forthe Hamiltonian h ( k ) = d τ z σ + d τ y σ z + d τ x σ z + d τ y σ , (D3)where d = m + 2 t cos( k x ) + 2 t cos( k y ) d = λ d = 2 λ sin( k x ) d = 2 t sin( k y ) . (D4)6The eigenvalues of h ( k ) read ± (cid:112) d + d + ( d ± d ) . For concreteness, we choose t = t = t = λ = 1 / m = 4 / λ to zero, as shown in Fig. 7(a), and the gap closing points sit at k = ( ± arccos( − √ ) , ± π/ ≈ ( ± . , ± . p m
1. When the gap is small but nonzero, the positions of valleys can bedetermined numerically by locating the peaks of Berry curvature (Fig. 7(b)), which are close to the gap closing points.Finally, based on the TB model Eq. (D3), we discuss the quantization and physical consequence of the VCN changeacross the λ = 0 gap closing. Without loss of generality, we take the valley in the k x,y > k x,y > Appendix E: (111) HgTe/CdTe Quantum Well
In this section, we provide more details on the analysis of the HgTe QW. Before going into details, we first introducesome basic properties of the QW. Both HgTe and CdTe have the standard zinc-blende structure, similar to most II-VIor III-V compound semiconductors. The crystallographic space group of both compounds is F ¯43 m (space groupNo. 216). In the QW, HgTe serves as a well while Hg − x Cd x Te serves as the barrier. Similar to early experimentaland theoretical studies (6, 47–50), we use x = 0 . d -induced PET jump for E = 0 To describe the TPQT, we project the 6-band Kane model onto the bases ( | E , + (cid:105) , | H , + (cid:105) , | E , −(cid:105) , | H , −(cid:105) ) viasecond order perturbation (49) and get the following 4-band model h (0) eff ( E = 0) = E + B k (cid:113) + Bk (cid:113) + m A k + − i A k + A k − − Bk (cid:113) − m − i A k +
00 i A k − Bk (cid:113) + m − A k − i A k − − A k + − Bk (cid:113) − m , (E1)where the values of the parameters are listed Tab. 4, k (cid:113) = k + k , k ± = k ± i k , and k and k are the momentaalong (1 , − ,
0) and (1 , , − k -linear term A due to the reduction of the full rotational symmetry to C rotation symmetry. In theEq. (E1), the TQPT shown in Fig. 2 (a) of the main text occurs at m = 0. To show the jump of the symmetry-allowedPET components at the gap closing, we need to introduce the electron-strain coupling h (1) eff based on the symmetry: h (1) eff = ξ u + ξ u ξ u − − i ξ u − ξ u + − ξ u i ξ u − − i ξ u + ξ u ξ u + i ξ u + ξ u − − ξ u , (E2)where u = u + u and u ± = u − u ± i( u + u ). This electron-strain coupling is in the most generalsymmetry-allowed form to the leading order of u ij , which definitely includes the IB terms, ξ and ξ . With Eq. (E1)and Eq. (E2), the independent PET component γ can be derived analytically as γ = − e sgn( m )( A ξ + A ξ )2 π ( A + A ) (cid:0) A + A − | Bm | + 2 Bm (cid:1) ( A + A + 4 Bm ) , (E3)7resulting in the PET jump as ∆ γ = − e A ξ + A ξ π ( A + A ) . (E4)Based on Eq. (E3) and ξ , , , = 1eV (comparable to those in Ref. (25)), we plot the γ of the function of the widthin Fig. 2 (b) of the main text, which shows a jump around d = 65˚A. E -induced PET jump for fixed d After including the electric field, we project the modified Kane model onto the bases ( | E , + (cid:105) , | H , + (cid:105) , | E , −(cid:105) , | H , −(cid:105) )via second order perturbation and get the following 4-band model h (0) eff = E + B k (cid:113) + Bk (cid:113) + m A k + + D k − − i D k − − i A k + − i D k − A k − + D k − Bk (cid:113) − m − i A k + + i D k − D k + i A k − − i D k Bk (cid:113) + m D k − A k − i A k − + i D k D k − − A k + − Bk (cid:113) − m . (E5)Compared with Eq. (E1), the above Hamiltonian has three extra IB terms D , , brought by the electric field. In fact,it is now in the most general symmetry-allowed form up to the second order of the momentum for the HgTe/CdTeQW along the (111) direction. In addition, the parameter m (mass term) can also be controlled by electric field. Inthe contrast to (001) QW, the constant ( k -independent) IB terms in Ref. (47) are forbidden in Eq. (E5) by the C symmetry. The E dependence of the parameters are shown in Tab. 5 for d = 62˚A. Since Eq. (E2) is in the most generalform, the electron-strain coupling for E (cid:54) = 0 still keeps the form of Eq. (E2). With Eq. (E5), Eq. (E2), the parameterexpression, and ξ , , , = 1eV comparable as those in Ref. (25), the PET jump can be calculated.
3. Projection of the Kane Model
With bases ( | Γ , (cid:105) , | Γ , − (cid:105) , | Γ , (cid:105) , | Γ , (cid:105) , | Γ , − (cid:105) , | Γ , − (cid:105) ), the 6-band Kane model that we use for the (111)quantum well without the electric field reads h Kane ( k ) = h Γ ( k ) T ( k ) T † ( k ) h Γ ( k ) , (E6)where k = ( k , k , k ) with k = − i ∂ x , h Γ ( k ) = (cid:16) E c + (cid:126) m (cid:2) (2 F + 1)( k + k ) + k (2 F + 1) k (cid:3)(cid:17) σ , T ( k ) = − √ k + P (cid:113) k P √ k − P − √ k + P (cid:113) k P √ k − P , (E7) h Γ ( k ) = U + V W (cid:102) W W † U − V (cid:102) W (cid:102) W † U − V − W (cid:102) W † − W † U + V , (E8) U = E v − (cid:126) m [( k + k ) γ + k γ k ], V = (cid:126) m [ − ( k + k ) γ +2 k γ k ], W = √ (cid:126) m [ − i √ k ( γ − γ )+ k − { k , γ + γ } ], (cid:102) W = √ (cid:126) m [ k − ( γ + 2 γ ) − i √ k + { k , γ − γ } ], m is the mass of the electron, and the IB effect is neglected. Theelectric field can be included by adding V e = − e E x (E9)8 E Av E Ac F A γ A γ A γ A P − .
303 0 4 . . . . E Bv E Bc F B γ B γ B γ B − .
399 0 . − .
063 2 . − .
046 0 . Table 3:
Values of parameters in Eq. (E6) for Hg . Cd . Te/HgTe/Hg . Cd . Te quantum well. The unites of E v , E c are eV, the unit of P is eV˚A, and other parameters are dimensionless (50, 73). d/ ˚A m /eV B /(eV ˚A ) m /eV B /(eV ˚A ) A /(eV ˚A) A /(eV ˚A)60.00 -0.006700 39.88 0.005600 60.45 3.595 0.124861.00 -0.007570 40.90 0.004370 61.46 3.582 0.123762.00 -0.008400 41.94 0.003200 62.51 3.569 0.122563.00 -0.009240 42.99 0.002040 63.56 3.555 0.121464.00 -0.01009 44.08 0.0008850 64.65 3.540 0.120465.00 -0.01084 45.14 -0.0001650 65.71 3.527 0.119366.00 -0.01159 46.25 -0.001210 66.82 3.514 0.118267.00 -0.01235 47.41 -0.002250 67.98 3.500 0.117268.00 -0.01307 48.54 -0.003230 69.12 3.486 0.116269.00 -0.01374 49.75 -0.004160 70.33 3.473 0.115270.00 -0.01442 50.98 -0.005080 71.56 3.459 0.1143 Table 4:
Parameter values for Eq. (E1) at various widths d .to Eq. (E6).Due to the spatial dependence of the parameters, we require the anti-commutation form of some k -dependentterms, such as { k , γ } , to keep the Hamiltonian hermitian (72). The quantum well considered has the structureHg . Cd . Te/HgTe/Hg . Cd . Te, leading to the x dependence of parameters X = E v,c , F, γ , , as X = X A , | x | < d X B , | x | > d . (E10)The numerical values of the parameters in Eq. (E6) are listed in Tab. 3.The effective models are derived according to Ref. (6 and 49). We first numerically obtain the wavefunctions of E1,H1, LH1, HH2, and HH3 bands at k = k = E = 0, and project the remaining terms to the bases to get a 10 × ×
10 Hamiltonian to the E1 and H1 bands with second order perturbation toget Eq. (E1) and Eq. (E5). Keeping terms up to k and E order, the values of the parameters are listed in Tab. 4 andTab. 5.
4. Construction of the Hamiltonian based on symmetry
As discussed in the main text, the symmetry group of interest is generated by the three-fold rotation C along (111),and the mirror m perpendicular to (1 , ¯1 ,
0) and the TR operation T . With the bases ( | E , + (cid:105) , | H , + (cid:105) , | E , −(cid:105) , | H , −(cid:105) ),9 m /eV B /(eV ˚A ) m /eV4182 ( − e E / (eV˚A − )) − − e E / (eV˚A − )) +41.94 0.003200 -17.21 ( − e E / (eV˚A − )) B /(eV ˚A ) A /(eV ˚A) A /(eV ˚A)534600 ( − e E / (eV˚A − )) +62.51 67320 ( − e E / (eV˚A − )) +3.569 293.0 ( − e E / (eV˚A − )) +0.1225 D /(eV ˚A ) D /(eV ˚A ) D /(eV ˚A)724.7 ( − e E / (eV˚A − )) -1947 ( − e E / (eV˚A − )) -1196 ( − e E / (eV˚A − )) Table 5:
Parameter values for Eq. (E5) for d = 62˚A. IR Expressions A , + σ τ , σ τ z , k + k , u + u A , − σ x τ − A , + A , − σ y τ − , σ z τ , σ z τ z E, + ( σ z τ y , σ τ x ), ( σ y τ y , σ x τ y ), (2 k k , k − k ), ( u + u , u − u ) E, − ( σ z τ x , − σ τ y ),( σ y τ x , σ x τ x ),( σ y τ + , − σ x τ + ), ( k , k ) Table 6:
The irreducible representations (IRs) of C v and TR symmetries. A , A and E are IRs of C v and ± areparity under TR. τ ± = ( τ ± τ z ) / C ˙= e − i π − i π
00 0 0 − m ˙= − i σ x τ = − i 00 0 0 − i − i 0 0 00 − i 0 0 T ˙= − i σ y τ = − −
11 0 0 00 1 0 0 K , (E11)where τ ’s and σ ’s are Pauli matrices for E , H indexes and ± indexes, respectively. According to the symmetryrepresentations, the matrix and momenta of the effective model can be classified as Tab. 6.From Tab. 6, the most general symmetry-allowed Hamiltonian without the electron-strain coupling can be derivedto the k order, resulting in Eq. (E5). As shown in Tab. 6, u ij behaves the same as the k term, and thereby theelectron-strain coupling has the same form as the k term in Eq. (E5). Appendix F: BaMnSb In this section, we include more details on BaMnSb .0 Fig 8:
The crystalline structure of BaMnSb generated from CrystalMaker.
1. Review
In this part, we review the form and the dispersion of the TB model derived in Ref. (52) for integrity. This partdoes not contain any original results. More details can be found in Ref. (52).According to the main text, there are two Sb atoms in one unit cell, labeled as 1 and 2, that have sub-lattice vectors τ = ( x a,
0) and τ = ( x a, b/ a, b are the lattice constants of the unit cell in x, y direction and thevalues of x , are given later. Combined with p x and p y orbitals, the bases of the TB model are | R + τ i , α, s (cid:105) with thelattice vector R = ( l x a, l y b ) ( l x,y ∈ Z ), the sublattice index i = 1 ,
2, the orbital index α = p x , p y , and the spin- z index s = ↑ , ↓ . The TB model consists of the on-site term H , the nearest-neighboring (NN) hopping H and the next-NNhopping H in the TB model, i.e. H T B = H + H + H . H has the form H = (cid:88) R ,i c † R + τ i M i c R + τ i (F1)with c † R + τ i = ( c † R + τ i ,p x , ↑ , c † R + τ i ,p x , ↓ , c † R + τ i ,p y , ↑ , c † R + τ i ,p y , ↓ ) . (F2) H reads H = (cid:88) R (cid:88) n =1 c † R +∆ R n + τ T n c R + τ + h.c. , (F3)where ∆ R = (0 , R = ( a, R = ( a, − b ) and ∆ R = (0 , − b ). H reads H = (cid:88) R ,i (cid:88) n = x,y c † R +∆ R n + τ i Q ni c R + τ i + h.c. , (F4)1where ∆ R x = ( a,
0) and ∆ R y = (0 , b ). The forms of M ’s, T ’s, and Q ’s are M = (cid:101) m τ σ + (cid:101) m τ z σ + λ τ y σ z ,M = C OS z M ( C OS z ) † ,T = t τ σ + t τ x σ + it τ y σ , T = τ z σ y T τ z σ y f ( α ) ,T = τ z σ y T τ z σ y , T = T f ( α ) ,Q x = t τ σ + t τ z σ , Q x = t τ σ + t τ z σ ,Q y = C OS z Q x ( C OS z ) † , Q y = C OS z Q x ( C OS z ) † , (F5)where f ( α ) = 0 . α + 1, and C OS z = ( − i τ y )e − i σz π is the representation of the four-fold rotation along z in the orbitaland spin subspace. α is the dimensionless distortion parameter; α = 0 and α = 1 correspond to the non-distortedand fully distorted cases, respectively. Moreover, the distortion effect on the relative atom positions is chosen as x = + (0 . − ) α and x = 0 . α , while we neglect distortion-induced change of a and b .The numerical calculation is done for (cid:101) m = 0 , (cid:101) m = 0 . , λ = 0 . eV , t = 1eV ,t = 2eV , t = 0 , t = 0 . , t = − . ,t = 0 . , t = − . , and a = b = 4 . . (F6)The energy dispersion for α = 1 is shown in the supplementary material of Ref. (52).
2. TB Calculation of PET
The main effect of the strain in the TB model is to change the hopping amplitudes among atoms (16, 74), whichcan be modeled by performing the following replacement (74) to the hopping parameters: t ab → (cid:18) − β δ i δ j u ij | δ | (cid:19) t ab (F7), where t ab is the hopping parameter between atoms at r a and r b in the non-deformed case, and δ = r a − r b . β is theelectron-phonon coupling parameter whose value for BaMnSb has not been determined, and thereby we adopt thetypical value β = 2 for the TMDs (74) to give a reasonable estimation of the PET jump.
3. Effective Model Analysis
To analytically demonstrate the PET jump, we project the tight-binding model into the subspace spanned by twodegenerate states at each gap closing point (valley). As discussed in Ref. (52), the resultant effective model reads h (0) ± ( q ) = ( E ± v q y ) τ ± v q y τ z ± v q x τ x ± ( E + λ ) τ y , (F8)where h ± ( q ) is around K ± with q = k − K ± , the two bases of h (0)+ ( h (0) − ) are | K + , p x ± i p y , ↑(cid:105) ( | K − , p x ± i p y , ↓(cid:105) ), andthe term with small coefficient has been omitted. E and v are given by the distortion, λ labels the SOC strength,and we choose E < , λ > E to E + λ = 0, which changes the Z index since only one Dirac cone appears inhalf 1BZ. To study the PET jump, we include the electron-strain coupling with the form h (1) ± ( q ) = N τ + N τ x + N τ z , (F9)where N = ξ xy ( u xy + u yx ) and N i = ξ i,xx u xx + ξ i,yy u yy for i = 0 ,
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