Robust zero-energy states in two-dimensional Su-Schrieffer-Heeger topological insulators
Zhang-Zhao Yang, An-Yang Guan, Wen-Jie Yang, Xin-Ye Zou, Jian-Chun Cheng
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Robust zero-energy states in two-dimensionalSu-Schrieffer-Heeger topological insulators
Zhang-Zhao Yang, An-Yang Guan, Wen-JieYang, Xin-Ye Zou,
1, 2, ∗ and Jian-Chun Cheng
1, 21
Key Laboratory of Modern Acoustics, MOE,Institute of Acoustics, Department of Physics,Collaborative Innovation Center of Advanced Microstructures,Nanjing University, Nanjing 210093, People’s Republic of China State Key Laboratory of Acoustics, Chinese Academy of Sciences,Beijing 100190, People’s Republic of China bstract The Su-Schrieffer-Heeger (SSH) model on a two-dimensional square lattice has been consid-ered as a significant platform for studying topological multipole insulators. However, due to thehighly-degenerate bulk energy bands protected by C v and chiral symmetry, the discussion of thezero-energy topological corner states and the corresponding physical realization have been rarelypresented. In this work, by tuning the hopping terms to break C v symmetry down to C v symme-try but with the topological phase invariant, we show that the degeneracies can be removed anda complete band gap can be opened, which provides robust protection for the spectrally isolatedzero-energy corner states. Meanwhile, we propose a rigorous acoustic crystalline insulator andtherefore these states can be observed directly. Our work reveals the topological properties of therobust zero-energy states, and provides a new way to explore novel topological phenomena. I. INTRODUCTION
The concept of higher-order topological (HOT) phases arising from the modern theoryof charge polarization in solids have greatly extended the class of the topological insulators(TIs) [1–8]. Distinct from the bulk-boundary correspondence of the traditional TIs [9–13]: the Chern insulators that require breaking time-reversal symmetry [14–27] and the Z insulators that are time-reversal invariant [28–44], these HOT insulators characterized byquantized nontrivial bulk polarization are protected by intrinsic crystalline symmetries andcan induce the filling anomalies of the fractional charges at the boundaries or corners ofthe crystal structures [3, 7, 45–47]. Recently, the HOT insulators have been theoreticallyand experimentally demonstrated in circuits [48, 49], microwaves [50], optics [51–55] andacoustics [56–62].Generally, the observation of the obstructed topological states induced by the chargefractionalization always requires spectral isolation, for that the robust corner states are atthe mid-gap if particle-hole symmetry or chiral symmetry is preserved [52]. Otherwise, thesetopological states may degenerate with the trivial bulk modes, and then act as symmetry-protected bound states in the continuum [63, 64]. Especially, for the lattices with C v symmetry as well as chiral symmetry, the double degenerate bands at the high symmetry ∗ [email protected] M and Γ always result in the gap being closed [65, 66], which naturally hinders theidentification of the zero-energy topological corner states. Previous works on C v -symmetricclassical systems have presented the domain-wall HOT states located in the lower gap thatis characterized by a dipole moment [55], but the discussion on the zero-energy corner statesin such the systems and the corresponding physical realization still need to be addressed.In this paper, we demonstrate the existence of the isolated zero-energy topological cornerstates in a two-dimensional (2D) Su-Schrieffer-Heeger (SSH) model, and present the corre-sponding realization in acoustic system. By judiciously breaking C v symmetry with thepreservation of all the other certain symmetries, i.e., C v symmetry and chiral symmetry,which ensure the topological phase invariant during the process, the degeneracy of the modesat the high symmetry points can be removed, and the zero-energy topological corner statescan exist at the mid-gap. Meanwhile, we show that these zero-energy states are so robustagainst perturbations even if all symmetries are reduced. According to these results, wethen numerically propose an acoustic HOT insulator which rigorously corresponds to thepresented 2D SSH model, and all the theoretical predictions can be directly observed.This paper is organized as follows. The 2D SSH theoretical model along with its topo-logical states is introduced in Sec. II. In Sec. III, the discussion of the robustness of thezero-energy topological corner states is presented. In section IV, the corresponding acousticHOT insulator is proposed. Finally, a summary is given in Sec. V. A few appendixes areprovided as a supplement to the discussion in the main text. II. TWO-DIMENSIONAL SSH MODEL WITH ZERO-ENERGY TOPOLOGICALSTATES
In this section, we present the topological properties of the 2D SSH model. As shownin Fig. 1(a), there are four atomic sites (labeled with 1-4, respectively) within the squarelattice. Here, we neglect the impact of the onsite energy and only consider the nearest-neighbor hopping. The corresponding first Brillouin zone (BZ) is depicted in Fig. 1(b), andthe Hamiltonian of this model can be given as h ( k ) = − ( w x + v x cos k x ) τ x σ + v x sin k x τ y σ z − ( w y + v y cos k y ) τ x σ x − v y sin k y τ x σ y , (1)3here k = ( k x , k y ) and { w i , v i , k i } ( i = x, y ) are the intra- and inter-lattice hopping terms,and the basis reciprocal vector in the i -direction, respectively. τ and σ are the Pauli matrix,while σ is the identity matrix.According to the theory of topological multipole insulators, the charges are predicted toaccumulate at the boundaries of the lattice (we label the left (right) boundary as x -edge,and the up (bottom) boundary as y -edge, respectively, due to the translation invariance), asthe total effect of the occupied bands acts as dipole moments in the corresponding direction.In 2D crystalline systems, the topological index can be characterized by the 2D Zak phasewhich is related to the charge polarization P = ( P x , P y ), where [1, 2] P j = − π ) Z BZ Tr[ A j, k ] d k , (2)where [ A j, k ] mn = − i h u m k | ∂ k j | u n k i ( j = x, y ) is the non-Abelian Berry connection (where | u m k i is the Bloch wave function and m , n run over the occupied bands). Note that the twocomponents P x and P y are independent with each other. On the other hand, when thelattice is C v -symmetric, which also implies chiral symmetry, there are always degeneratebands existing at the zero energy of the high symmetry points M and Γ of the BZ. Tolift the degenerates, we then introduce a set of basis parameters { w, v, d } and redefinethe hopping terms as w x = w (1 − d ), v x = v (1 − d ) and w y = w (1 + d ), v y = v (1 + d ),respectively. Therefore, it is obvious to see that the charge polarization P is yielded between(0 ,
0) and (1 / , / β = v/w ; meanwhile,the nonzero d ∈ ( − ,
1) results in C v symmetry broken down to C v symmetry. As a result,if only one occupied band is considered, β > C v symmetry yield the polarization P = (1 / , / d > /β indicates that the degenerates are totally removed and a complete band gap isopened at zero energy. Figure 1(d) shows the bulk energy band structure when { v, β, d } = { , , . } . As a comparison, the energy band structure when d = 0 is presented in Fig.1(c). Further, we show the existence of the topological edge states and zero-energy cornerstates in this model.As discussed above, the topological edge-localized states are predicted to emerge alongwith the dipole moments. To explain this, we open the boundaries in the y ( x ) direction whileremaining the boundaries in the x ( y ) direction periodic, and the corresponding quasi-one-dimensional tubular structures are illustrated in Figs. 2(a) and 2(b), respectively. Figures4(c) and 2(d) show the energy band structures of the ribbon-shaped superlattices in Figs.2(a) and 2(b), respectively. It can be seen that in both sides, the bands with edge-localizedstates are spectrally isolated from the bulk bands. It is worth noting that the dipole momentsin this model are actually hosted by the first and fourth bands of the lattice, whereas thetotal polarization vanishes in the middle gap, for that there are two occupied bands belowthis gap. Therefore, the fact that the edge states at y -edges are located in the middle gap isdue to C symmetry broken that leads to the large offset of the bulk bands. Note that thesame results can also be characterized by the Wannier bands as illustrated in Appendix. A.Although the total polarization in the middle gap vanishes, the independent componentsof the dipole moments still allow us to define a quadrupole index as [67] Q xy = occ X n =1 P nx P ny . (3)For that there are two occupied bands for the middle gap, Q xy = 1 / ×
15 lattices, and thecorresponding calculated energy spectrum is presented in Fig. 3(b). It can be seen that inaddition to the y -edge-localized states, there are four degenerate zero-energy corner statesprotected by chiral symmetry emerging in the middle gap. The corresponding spatial energydistributions of the four degenerate corner states are presented in Fig. 3(c). Meanwhile, dueto the fact that these corner states are isolated from the trivial bulk modes, all these statescan carry 1 / III. ROBUSTNESS OF THE ZERO-ENERGY CORNER STATES
In this section, we present the high robustness of these isolated states. As discussedabove, apart from time-reversal symmetry, the 2D SSH model also possesses C v symmetry,which includes mirror symmetries M x and M y , and chiral symmetry Π. Here, we add asmall perturbation h Λ per consisting of random hopping terms to the bulk lattices of the finitestructure [Fig. 3(b)] as h Λ ( k ) = h ( k ) + h Λ per , (4)5here Λ = { M x , M y , Π } indicates that the perturbation respects the corresponding symme-try as [ h Λ per , Λ] = 0. As a result, these well-tuned symmetries are broken down to only specificindicated symmetries [52, 63]. In Figs. 4(a)-4(d) we show the energy spectra along with thedensity functions of the perturbed structures, and it is obvious to see that the zero-energystates are so robust against perturbations which breaks mirror symmetries. In particular,even if all symmetries are broken (including chiral symmetry being slightly reduced), thetopological corner states are still almost pinned to the zero energy and not degenerate withthe bulk modes [Fig. 4(d)].We argue that the robust zero-energy states in the present 2D SSH model are actuallyprotected by both the band gap and chiral symmetry [63]. Whereas, once the introducedperturbations are strong enough to impact the dipole moments, these corner states maybe unpinned and shifted to degenerate with the trivial bulk modes. In particular, thelarge distortion at the corners may result in the double-projections being removed, and theprojection states which originate from the single dipole moment can be separated from theedge states and isolated to the corners. To explain this, the field distributions of the isolatedand degenerate zero-energy states are shown in Figs. 4(e) and 4(f), while the single-dipole-projected states are presented in Figs. 4(g) and 4(h), respectively.
IV. ACOUSTIC REALIZATION OF THE 2D SSH MODEL
The proceeding discussion has theoretically demonstrated the existence of robust zero-energy topological corner states isolated in the mid-gap. In this section, we propose arigorous physical model based on acoustic resonance system to observe these topologicalstates directly. Figure 5(a) provides the schematic of the crystalline structure that spans10 ×
10 lattice according to the model presented in Fig. 3(b). As illustrated, there are fouridentical cubic acoustic cavities connected by waveguide tubes in one lattice. The side lengthof the cavity is s = 6 cm, and the radius of the tube is r = 0.5 cm. For the bulk lattices,the lengths of the intra- and inter-lattices tubes in the two directions are l wx = 15.8 cm, l wy = 2.7 cm, l vx = 4.7 cm and l vx = 0.3 cm, respectively; for the lattices at the boundaries,all the outermost tubes need a correction length of 0 . r [68]. The mass density of the airand the corresponding sound speed are ρ = 1.29 kg / m and c = 343 m/s, respectively.Therefore, the bulk lattice of the acoustic structure then can be rigorously described by the6heoretical model given in Eq. (1). The corresponding hopping terms of the Hamiltoniancan be calculated as w x = 2 . × , w y = 1 . × and β = 3, respectively, and the on-siteterms can be obtained as ω = w x + w y + v x + v y (see detailed derivation in Appendix B).Accordingly, the numerical and theoretical results of the bulk energy bands are shown inFig. 5(b), and the corresponding four eigenstates at the high symmetry points X and Y ofthe acoustic structure are presented in Figs. 5(c) and 5(d), respectively.Further, we apply soft boundary condition to the outermost tubes of the finite structure[68, 69], and the calculated energy band structures of the two ribbon-shaped superlatticesconsisting of 15 bulk lattices in the x -direction and the y -direction are presented in Figs. 6(a)and 6(b), respectively, which corresponds to the theoretical results shown in Figs. 2(c) and2(d). For the finite structure with open boundaries [Fig. 5(a)], the calculated eigenfrequencyspectrum is then presented in Fig. 6(c). To verify the existence of acoustic higher-ordertopological states, the sound pressure distributions of the bulk state, edge state and cornerstate are provided in Fig 6(d). Such the results are perfectly corresponding to the theoreticalpredictions. In particular, the frequency of the acoustic corner states is 393 Hz, which isconsistent with the theoretical prediction as f = ω / π = 385 Hz. V. SUMMARY
In summary, we have demonstrated the existence of the robust zero-energy topologicalcorner states in the 2D SSH model, and directly observed these states based on a judiciouslydesigned acoustic crystalline insulator. By breaking C v symmetry down to C v symmetrywithout impacting the topological properties of the bulk, the degeneracy of the bulk bandscan be continuously removed, and result in a complete band gap opened at zero energy.Meanwhile, we show that with the protection of the band gap, these zero-energy corner statescan be exponentially confined to the corners and are robustly against random perturbationsintroduced within the system. We also propose a rigorous acoustic topological crystallineinsulator to verify these states directly. Our findings are expected to not only be helpfulfor the understanding of HOT states in topological multipole insulators, but also provide aplatform for the design and potential applications of topological materials.7 CKNOWLEDGMENTS
Z.-Z. Y thanks S.-P. Song for helpful discussions. This work was supported by the Na-tional Key R&D Program of China (Grant No. 2017YFA0303700), National Natural ScienceFoundation of China (Grant Nos. 11634006, 11934009, and 12074184), the Natural ScienceFoundation of Jiangsu Province (Grant No. BK20191245), State Key Laboratory of Acous-tics, Chinese Academy of Sciences.
Appendix A: WANNIER BANDS OF THE 2D SSH MODEL
We briefly introduce the Wilson-loop method and the corresponding results of the 2DSSH model. Here, we consider two occupied bands, and the discretized Berry connectionmatrix in the x and y directions can be defined as [ F i, k ] mn = h u m k +∆ k i | u n k i ( i = x, y ), where∆ k i is a small positive quantity of k in the i -direction. The corresponding Wilson-loopoperator then can be defined as W i, k = Π N − t i =0 F i, k + t ∆ k i , where t i = 2 π/ ∆ k i is the number ofthe points that the loop in the i direction is discretized into. With fully periodic boundaryconditions, W x, k can be diagonalized as W x, k = X j | ν jx, k i e i πν jx ( k y ) h ν ji, k | , (A1)where | ν jx, k i is the eigenstates and ν jx ( k y ) gives the Wannier centers of the two occupied bandsin the k x -direction [2]. By traversing the loop over the entire BZ along the k y -direction, wethen can obtain two Wannier bands that are related to the polarization of the Bloch bands.The Wilson-loop operators in the k y -direction W y, k can be calculated similarly, yieldingthe Wannier bands as ν jy ( k x ). For a single occupied band, the quantized Wannier bandsof the 2D SSH model are presented in Figs. 7(a) and 7(b), respectively, which representsthe existence of dipole moments on both x -edges and y -edges. The polarizations of the twooccupied bands case are depicted in Figs. 7(c) and 7(d), respectively. It is obvious to seethat the two Wannier bands are quantized to be zero for the loops in the k x -direction, andthe Wannier bands in the k y -direction are 1/2. This result implies that the charges can stillaccumulate on the y -edges, but disappear on the x -edges, which exactly corresponds to theresults shown in Figs. 2(c) and 2(d), respectively.In addition, even if the total bulk polarization of the two occupied bands vanishes, the8harges can still accumulate on the corners [2]. This process can be discribed by the nestedWilson loops. Due to the fact that the Wannier bands are gapped in this model, the twobands can be labeled with “ ± ”, respectively. The Wannier band subspaces then can bedefined as | w ± i, k i = occ X n =1 | u n k i [ ν ± i, k ] n . (A2)The nested discretized Berry connection then can be defined as F ± i, k = h w ± k +∆ k i | w ± k i , and thecorresponding nested Wilson loops can be obtained as ˜ W ± i, k = Π N − t i =0 F ± i, k + t ∆ k i . Finally, theassociated polarizations can be obtained as p ν ± x y = − i π Log[ ˜ W ± y,k x ] , (A3)and the same for p ν ± y x . Appendix B: DETAILED DERIVATION OF THE ACOUSTIC CRYSTALLINEMODEL
In this section, we show the derivation of the acoustic model [Fig. 5(a)]. Accordingto the schematic dipicted in Fig. 1(a), we define the impedances between the intra- andinter-lattice cavities in the two directions as Z xw = iωL xw , Z yw = iωL yw , Z xv = iωL xv and Z yv = iωL yv , respectively, and the intrinsic impedance of the cavities as Z c = 1 /iωC . Here, ω = 2 πf is the frequency of the wave, L qp is the acoustic mass of the corresponding tube,and C is the acoustic capacitance of the cavity. We then introduce a Bloch wave function | u i = [ u , u , u , u ] T into the bulk lattice. Therefore, for the specific site,e.g., the 1-site, theflow of the wave component can be described as [68, 69] − u Z c = u − u x +3 Z xw + u − u y − Z yw + u − u x − Z xv + u − u y +4 Z yv , (B1)where the superscript “ x + ” of the wave function indicates the nearest-neighbor site of 1-sitein the positive x -direction, and so on. For the other sites, we can obtain the similar forms.Further, we define w = − ω Z c /Z w and v = − ω Z c /Z v . After Fourier expansion, the flowsof | u i in the entire bulk lattice can be written in the reciprocal space as H ( k ) | u i = ω | u i , (B2)9here H ( k ) = h ( k ) + ω τ σ , (B3)where ω = w x + w y + v x + v y determines the frequency of the zero-energy states.For the present acoustic resonance model, we can calculate all the parameters of the bulklattice as L w = ρ ( l w + 1 . r ) πr ,L v = ρ ( l v + 1 . r ) /πr ,C = a /ρc . (B4)Note that for the boundary lattices, due to symmetry broken, corrections should be addedas stated in the main text. As a result, the hopping terms as well as the on-site terms canbe directly obtained. [1] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Quantized electric multipole insulators,Science , 61 (2017).[2] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Electric multipole moments,topological multipole moment pumping, and chiral hinge states in crystalline insulators,Phys. Rev. B , 245115 (2017).[3] L. Fu, Topological crystalline insulators, Phys. Rev. Lett. , 106802 (2011).[4] J. Langbehn, Y. Peng, L. Trifunovic, F. von Oppen, and P. W. Brouwer,Reflection-symmetric second-order topological insulators and superconductors,Phys. Rev. Lett. , 246401 (2017).[5] Z. Song, Z. Fang, and C. Fang, ( d − , 246402 (2017).[6] F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig, andT. Neupert, Higher-order topological insulators, Sci. Adv. , eaat0346 (2018).[7] W. A. Benalcazar, T. Li, and T. L. Hughes, Quantization of fractional corner charge in C n -symmetric higher-order topological crystalline insulators, Phys. Rev. B , 245151 (2019).[8] F. Schindler, M. Brzezi´nska, W. A. Benalcazar, M. Iraola, A. Bouhon, S. S. Tsirkin,M. G. Vergniory, and T. Neupert, Fractional corner charges in spin-orbit coupled crystals,Phys. Rev. Research , 033074 (2019).
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ΓY XM M0123-1-2-3 E n e r gy d = 0 ΓY XM M0123-1-2-3 E n e r gy d = 0.65 (a) (b)(c) (d) FIG. 1. (a) Schematic of the 2D SSH model. (b) First BZ of the square lattice. Energy bandstructures of the bulk lattice when (c) d = 0 and (d) d = 0 . y y -edge y -edge y x x -edge x -edge E n e r gy k x (2 π ) 0123-1-2-3 E n e r gy k y (2 π ) (a) (b)(c) (d) FIG. 2. Schematics of tubular structures with (a) closed x -edges and open y -edges, and (b) open x -edges and closed y -edges. (c) Energy band structure of the ribbon-shaped superlattice in (a).(d) Energy band structure of the ribbon-shaped supperlattice in (b). x P y P Q xy E n e r gy Bulk state y -edge stateCorner state Number of states (a) (b)(c)
FIG. 3. (a) Illustration of the polarizations in the 2D SSH model. (b) Energy spectrum of the15 ×
15 finite structure. (c) Spatial energy distributions of the four degenerate zero-energy cornerstates.
300 600 900 0.2 0.4 0.6 E n e r gy Number of states Density of states [a.u.]
Mx and Chiral E n e r gy Number of states Density of states [a.u.]
My and Chiral E n e r gy Number of states Density of states [a.u.]
Chiral E n e r gy Number of states Density of states [a.u.]
None
Isolation Degenerate x -edge projection y -edge projection (a) (b)(c) (d)(e) (f) (g) (h) FIG. 4. Energy spectra along with the corresponding denity functions of the perturbed structureswith (a) M x symmetry and chiral symmetry, (b) M y symmetry and chiral symmetry, (c) only chiralsymmetry and (d) none symmetry. Insets: Schematics of a lattice with distorted hopping terms oron-site terms. Spatial energy distributions of (e) isolated zero-energy corner states, (f) degeneratezero-energy corner states, (g) corner states projected by x -edge dipole moments and (d) cornerstates projected by y -edge dipole moments. x rl xw l yw s l yv l yw Bulk lattice
ΓY XM M F r e qu e n c y [ H z ] TheorySimulation p x s d p y p y d s p x (a) (b)(c)(d) FIG. 5. (a) Schematic of the finite acoustic crystalline insulator spanning 10 ×
10 lattices. Inset:Bulk lattice of the acoustic structure. (b) Energy band structures of the acoustic bulk latticecalculated by simulation (blue lines) and theory (red dots), respectively. Four eigenstates from thebottom band to the top band of the high symmetry points (c) X and (d) Y , respectively. F r e qu e n c y [ H z ] k x (2 π ) F r e qu e n c y [ H z ] k y (2 π ) F r e qu e n c y [ H z ] Number of state
MIN MAX
BulkEdgeCorner (a) (b)(c) (d)
FIG. 6. Energy band structures of the acoustic ribbon-shaped superlattices with (a) periodic x -edges and open y -edges, and (b) open x -edges and periodic y -edges. Both the two superlatticeshost topological edge states (isolated red lines). (c) Numerically calculated eigenfrequency spetrumof the acoustic structure in Fig. 5(a). (d) Sound pressure distributions of the bulk state, y -edgestate and zero-energy corner state. k x (2 π ) 0.5-0.5 p y k y (2 π ) 0.5-0.5 p x k y (2 π ) 0.5-0.5 p x k x (2 π ) 0.5-0.5 p y (a) (b)(c) (d) FIG. 7. Polarizations of the 2D SSH model. (a)-(b) Wannier bands of one single occupied bandconsidered in the k x -direction and k y -direction, respectively. (c)-(d) Wannier bands of two occupiedbands considered in the k x -direction and k y -direction, respectively. Both the total polarizationsare zero.-direction, respectively. Both the total polarizationsare zero.