Higher-order topological phases in tunable C_3-symmetric photonic crystals
Hai-Xiao Wang, Li Liang, Bin Jiang, Junhui Hu, Xiancong Lu, Jian-Hua Jiang
HHigher-order topological phases in tunable C -symmetric photonic crystals Hai-Xiao Wang, ∗ Li Liang, Bin Jiang, Junhui Hu, Xiancong Lu, and Jian-Hua Jiang † School of Physical Science and Technology, Guangxi Normal University, Guilin 541004, China School of Physical Science and Technology, & Collaborative Innovation Center of Suzhou Nano Science and Technology,Soochow University, 1 Shizi Street, Suzhou 215006, China Department of Physics, Xiamen University, Xiamen 361005, China (Dated: February 24, 2021)We demonstrate that multiple higher-order topological transitions can be triggered via the con-tinuous change of the geometry in kagome photonic crystals composed of three dielectric rods. Bytuning a single geometry parameter, the photonic corner and edge states emerge or disappear withthe higher-order topological transitions. Two distinct higher-order topological insulator phases anda normal insulator phase are revealed. Their topological indices are obtained from symmetry rep-resentations. A photonic analog of fractional corner charge is introduced to distinguish the twohigher-order topological insulator phases. Our predictions can be readily realized and verified inconfigurable dielectric photonic crystals.
I. INTRODUCTION
Topological phases and phase transitions have been ex-tensively studied in electronic , photonic and acous-tic systems in the past decades. Recently, a newclass of topological insulators, higher-order topologicalinsulators (HOTIs), which are characterized by higher-order bulk-boundary (e.g., bulk-corner or bulk-hinge)correspondence, were discovered . Prototype HO-TIs include quadrupole and octupole topological insu-lators , three-dimensional HOTIs in electronicsystems with topological hinge states , and HOTIswith quantized Wannier centers . For example,quadrupole topological insulators, featured with a frac-tional bulk quadrupole moment, host gapped edge stateswith fractional dipole polarization and in-gap cornerstates accompanying a fractional corner charge. Anotherexample is the C -symmetric two-dimensional (2D) crys-tals which exhibit quantized bulk polarization, gappededge sates and corner states . In these and othertypes of HOTIs, the crystalline symmetry plays a keyrole in the underlying physics. HOTIs set up exampleswith multidimensional topological physics going beyondthe bulk-edge correspondence in conventional topologicalinsulators and semimetals and thus offer novel applica-tions in photonics and phononics.Despite the extensive studies on HOTIs, 2D photoniccrystals (PhCs) with C rotation symmetries are rarelystudied . In particular, such PhCs have rich higher-order topological phenomena which have not yet been re-vealed. Here, we show that by moving the dielectric rodscontinuously, the C -symmetric PhCs can switch betweentriangule and kagome lattice configurations, leading torich higher-order topological phases and phase transi-tions. Accompanying with such phase transitions, thecorner and edge states emerge or disappear, while thecorner charge changes between 0 and . The topologicalindices for various phases are deduced from the symme-try indicators which are closely related to the fractionalcorner charge . We also discuss the physical meaning റ𝑎 റ𝑎 𝑙 𝑎 𝑑 kagome Ⅰtriangular Ⅰ triangular Ⅱtriangular Ⅲkagome IⅡ kagome II 𝑥𝑦 d= d= l d=ld= l d= l d= l (b) (c)(f) (e) (d)(a) (g) 𝛼𝛽 𝛾 FIG. 1. (Color online) Geometric transitions in the 2D PhCswith C symmetry. The primitive cells are indicated by hexag-onal dotted lines with the lattice constant a and the sidelength l . A tunable parameter d with a range of 0 to 3 l (theparameter d is modulo 3 l ) is employed to illustrate the geo-metric transitions between triangular, kagome and breathingkagome configurations. By tuning the geometric parameter d , the C symmetry is preserved, while various configurationscan be generated, including (a) triangular I with d = 0, (b)kagome I with d = 0 . l , (c) triangular II with d = l , (d)kagome II with d = 1 . l , (e) triangular III with d = 2 l ,(f) kagome III with d = 2 . l . Each primitive cell consistsof three dielectric rods (possibly overlaping with each other)with identical radii r = 0 . a and permittivity (cid:15) = 15. of the fractional corner charge in the photonic context.The richness of the higher-order topological phases andtheir evolutions provide intriguing photonic phenomenaand potential applications in topological photonics whichcan be readily realized in genuine materials. a r X i v : . [ phy s i c s . op ti c s ] F e b Γ M K F r e qu e n c y ( c / a ) Γ MK Γ M K Γ (a) (b) d/l F r e qu e n c y ( c / a ) χ=[0,0] χ=[-1,0]χ=[-1,1] z )π0.250 2.75 3 (c)(d) χ=[0,0] triangular Ⅰ / Ⅱ / Ⅲ kagome Ⅰ / Ⅱ / Ⅲ breathing kagome 𝛼 𝛽 𝛾 𝛼 FIG. 2. (Color online) Photonic band structures of 2D PhCswith C symmetry for (a) d = 0 , l, l (i.e., triangular I, IIand III lattices), (b) d = 0 . l, . l, . l (i.e., kagome I, II andIII lattices), (c) d = 0 . l, . l, . l (i.e., breathing kagomelattices). (d) The eigen-frequencies of the first and secondphotonic bands at the K point as functions of d . Band gapsof distinct topology are painted with different colors. Thetopological index χ is labeled for each region. The Wanniercenter for each region is depicted as well. Insets illustratethe phase distributions of the eigenstates of the first photonicband at the K point for various d ’s. II. HIGHER-ORDER TOPOLOGICAL PHASESIN TUNABLE C -SYMMETRIC PHOTONICCRYSTALS We study 2D hexagonal PhCs of C rotation symmetryas illustrated in Fig. 1. The lattice vectors are denotedas (cid:126)a = ( a,
0) and (cid:126)a = ( a , √ a ) where a is the latticeconstant. The side length of the unit cell is denoted as l = a/ √
3. The simplest configuration is the triangularlattice with a dielectric rod at the center of each unit-cell [Fig. 1(a)] which can be regarded as the special casewhere three identical dielectric rods overlap with eachother. By moving the three dielectric rods along the threesymmetry lines, as indicated by the arrows in Fig. 1(b),the PhC undergoes a continuous geometry transforma-tion which includes three triangular lattice configurations(denoted as triangular I, II and III) and three kagomelattice configurations (denoted as kagome I, II and III)[see Figs. 1(b)-1(f)]. The configurations between thesesix special cases are the breathing kagome lattices. Thewhole cycle of the continuous deformation encompasses d from 0 to 3 l (see Fig. 1). The kagome lattices arecharacterized by d = ( n + ) l with n = 0 , ,
2, whilethe triangular lattices are characterized by d = nl with n = 0 , , α ) or the corner of the unit-cell ( β or γ )[see Fig. 1(g)]. Unlike the positions of the dielectric rods,the Wannier center is constrained by the crystalline sym-metry and thus cannot change continuously. The changeof the Wannier center is nonadiabatic which has to beachieved by closing and reopening of the band gap, asshown in details below.We first provide the photonic band structures for nineprototype cases in Figs. 2(a-c) where we use c/a asthe frequency unit ( c is the speed of light in vacuum).Throughout this work, we focus on the low-lying pho-tonic bands due to the transverse magnetic (TM) har-monic modes. All the numerical simulation results arecarried out by using the finite element numerical solverCOMSOL Multiphysics. The photonic bands for the tri-angular I, II and III configurations are shown in Fig. 2(a)which indicates that the three triangular configurationshave identical band structure. This phenomenon is be-cause the three triangular configurations differ only by apartial lattice translation, while the structure of the 2Darray of the dielectric rods remain the same. However,the symmetry representations of the photonic bands aredistinct for the three triangular configurations. Hence,their topological properties are different. In particular,the location of the Wannier center is distinct for the threetriangular configurations, as revealed below.Similarly, the photonic band structures for the kagomeI, II and III configurations are identical because they canbe related to each other by partial lattice translations[Fig. 2(b)]. Such translations change the location of theWannier center as well as the symmetry representationsof the Bloch bands and their topological properties.Furthermore, as shown in Fig. 2(d), the photonic bandstructure is identical, if two configurations differe by aninteger times of l in the geometry parameter d . Sincethe lattice periodicity for the translation along the blackarrows in Fig. 1(b) is 3 l , there are three different config-urations with the same photonic band structure, wherethe geometry parameter d differs by an integer times of l . The translation of l along the black arrows in Fig. 1(b)shifts the unit-cell center to the unit-cell corner withoutchanging the pattern of the 2D array of the dielectricrods. The photonic band structure is insensitive to suchglobal shifts. Therefore, configurations differ by an inte-ger times of l in the geometry parameter d have the samephotonic band structure. To further demonstrate such aperiodicity of the photonic band structure, we presentthe photonic bands for three breathing kagome configu-rations with d = 0 . l , 1 . l , and d = 2 . l in Fig. 2(c).It is seen that their photonic band structures are identicalwith each other.The evolution of the first two photonic bands at the K point [i.e. k = ( π , d is systematically summarized in Fig. 2(d). Thereare three topologically distinct photonic band gaps (re-gions painted by different colors) which are characterizedby three different locations of the Wannier center, as in-dicated in the figure. The band gap between the firsttwo bands experiences closing and reopening during thechange of the parameter d . We find that the band gapcloses at the kagome I, II and III configurations where d = ( n + ) l with n = 0 , ,
2, separately. These three d ’s separate the whole region d ∈ [0 , l ] (bearing in mindthat the parameter d is modulo 3 l , since 3 l correspondsto a lattice periodic translation) into three topologicallydistinct phases: d ∈ ( − . l, . l ) where the Wannier cen-ter is at α , d ∈ (0 . l, . l ) where the Wannier center is at β , d ∈ (1 . l, . l ) where the Wannier center is at γ .The symmetry representation of the first photonicband at the K point is depicted by the phase profile ofthe electric field E z which are shown in Fig. 2(d) for sev-eral d ’s. We find that the C symmetry eigenvalue doesnot change within the same phase. Upon the topologicalphase transitions, i.e., the band gap closing and reopen-ing, the symmetry eigenvalue changes abruptly. We usethe symmetry indicators to characterize the bulk bandtopology. Following Ref. 43, the topological crystallineindex can be expressed by the full set of the C eigen-values at the high-symmetry points (HSPs). For a HSPdenoted by the symbol Π, the C eigenvalues can only beΠ n = e π ( n − / with n = 1 , ,
3. Here, the HSPs includethe Γ, K and K (cid:48) points. The full set of C eigenvalues atthe HSPs are redundant due to the time-reversal symme-try and the conservation of the number of bands belowthe band gap. The minimum set of indices that describethe band topology are given by ,[ K ] n = K n − n , n = 1 , K n ( n ) is the number of bands below theband gap with the C symmetry eigenvalue K n (Γ n ) atthe K (Γ) point. In this scheme, the Γ point is takenas the reference point to get rid of the redundance. Forthe trivial atomic insulators (i.e., band gap formed byuncoupled atoms), all the HSPs have exactly the samesymmetry eigenvalues. Therefore, the trivial atomic in-sulators have [Π n ] = 0 for all the HSPs. In constrast,any nonzero [Π n ] indicates a topological band gap thatis adiabatically disconnected from the trivial atomic in-sulator.For the C -symmetric PhCs, the topological indicescan be written in a compact form as χ = [[ K ] , [ K ]] . (2)We find that for all d ’s = 1 and = 0. Fur-thermore, from the C eigenvalue at the K point asindicated in Fig. 2(d), we find that the topological in-dex for the three parameter regions are: χ = [0 ,
0] for d ∈ ( − . l, . l ), χ = [ − ,
1] for d ∈ (0 . l, . l ), and χ = [ − ,
0] for d ∈ (1 . l, . l ). III. EMERGENCE AND EVOLUTION OF THECORNER AND EDGE STATES
Both the phase with χ = [ − ,
1] and that with χ =[ − ,
0] are higher-order topological phases that hosting gapped edge states and in-gap corner states. In con-trast, the phase with χ = [0 ,
0] is the trivial phase. Todemonstrate the higher-order topology, we construct alarge triangular supercell which is schematically shownin Fig. 3(a). In the supercell, the inside is the phase thatwe study, while the outside is the trivial band gap phasewith d = 0 . l . The side length of the supercell is 10 a ,while the inside structure has a side length of 4 a . Thewhole structure is surrounded by the PEC (i.e., perfectelectric conductor) boundary condition which is physi-cally a hard-wall boundary for photons.We study the evolution of the edge and corner stateswhen the parameter d of the inside PhC structure goesfrom 0 to 3 l . A number of prototype geometry is shownin Fig. 3(a). The results are presented systematically inFig. 3(b). Fig. 3(c) gives the electric field | E z | distri-butions of the eigenstates. Throughout this paper, theelectric field patterns of the corner states are given by thesuperpostion of | E z | of the three degenerate corner states.From the figure, it is seen that the edge and corner statesemerge only in the region with 0 . l < d < . l , i.e., thetwo higher-order topological phases. In particular, in theregion with 0 . l < d < . l , there are two types of edgestates emerging, as revealed previously in Ref. 41: type-Icorner states [denoted by the purple curve in Fig. 3(b),two examples (“A” and “C”) are shown in Fig. 3(c)]due to the nearest neighbor couplings and type-II cornerstates [denoted as the blue curve in Fig. 3(b), two exam-ples (“B” and “D”) are shown in Fig. 3(c)] due to long-range couplings. As the common band gap between theinside and outside structures become smaller, at d = l ,the type-II corner states are much less localized and be-come edge states alike [see “D” in Fig. 3(c)]. In con-trast, the type-I corner states remain well-localized anddistinguishable from the edge states [denoted as greenin Fig. 3(b)] and bulk states [denoted as green-gray inFig. 3(b)]. This indicates that the type-I corner statesare due to the bulk topology, while the type-II cornerstates may originate from long range couplings betweenthe adjacent edge states.The region with 1 . l < d < . l has not yet been stud-ied in the literature. We find that in this region thetype-II corner states are hardly seen. Meanwhile, thereare two sets of type-I corner states. Each set has threedegenerate corner states. One set has frequency higherthan the edge states, while the other set has frequencylower than the edge states. In both sets, the wavefunc-tions of the corner states are well-localized around thecorners, distinghuisable from the edge states (the greenband) and the bulk states (the green-gray bands). Ex-amples of the corner and edge wavefunctions at d = 1 . l and d = 2 . l are shown in Fig. 3(c). In some cases, forinstance, d = 2 . l , type-II corner states can be found.However, the wavefunctions are not well-localized at thecorners.We now explore the corner and edge states in anothertype of supercell. We design the supercell in such a waythat the inner structure is a PhC with the parameter d (a)(b)(c) MaxMin|E z | d =1.9 l E F G d =0.75 l A B d = l C D d =2.25 l I J H F r e qu e n c y ( c / a ) d/l F EG HJIAB CD d =0.75 l d =1.25 l d =1.75 l d =2.25 l FIG. 3. (Color online) (a) Schematic illustration of the large triangular supercells with two types of PhCs. The outer PhC has d = 0 . l , while the inner PhC has variable d . Several cases with different d ’s are shown in (a). (b) Eigen-frequencies of thephotons as functions of the geometry parameter d . The green-gray regions represent the bulk states, the green regions representthe edge states, while the purple and blue curves represent the type-I and type-II corner states, respectively. (c) Electric fieldpatterns of corner states and edge states with different d . Throughout this paper, the electric field patterns of the corner statesare given by the superpostion of | E z | of the three degenerate corner states. In the calculation, the side length of the supercellis 10 a , while the inside structure has a side length of 4 a . while the outer structure is the PhC with the parameter3 l − d . That is, we consider the edge and corner bound-aries between complimentary PhC structures. Such a su-percell architecture will induce intriguing edge and cor-ner boundaries [e.g., various zigzag edge boundaries asdepicted schematically in Fig. 4(a)]. The evolution ofthe bulk, edge and corner states are systematically sum-marized in Fig. 4(b) and 4(c).In this type of supercells, the edge and corner statesemerge only in the two topological regions, 0 . l < d < . l and 1 . l < d < . l , as shown in Fig. 4(b). In theregion with 0 . l < d < . l , the inner PhC has the topo-logical index χ = [ − ,
1] while the outer PhC has thetopological index χ = [ − , . l < d < . l , the topological indices of the inner andouter PhCs switch, i.e., the inner PhC has χ = [ − , χ = [ − , and the Wannier center picture.From Fig. 4(b), the bulk band gap closing is clearlyseen at the phase transition points, d = 0 . l , 1 . l and 2 . l .Type-II corner states can be found only in the higher-order phase with χ = [ − ,
1] (i.e., the phase studied inRef. 41; here 0 . l < d < . l ), but not in the higher-orderphase with χ = [ − ,
0] (i.e., 1 . l < d < . l ). For all casesin the region 0 . l < d < . l , the edge states are clearlyvisible [see Fig. 4(c)]. The bandwidth of the edge statesis considerably larger in this type of supercells comparedwith the supercell studied in Fig. 3. As a consequence,the corner states are less localized, particularly in theregion with 1 . l < d < . l where the corner states livein the small band gap between the edge and bulk states. IV. FRACTIONAL CORNER CHARGE
We now show that the higher-order band topology canalso be manifested in the fractional corner charge. Eventhough we are considering photonic bands and photonicstates in this work, it is possible to define an analog F r e qu e n c y ( c / a ) (b) MaxMin |E z | (c) HG IJAB DEFC d/l
G H d =2 l JI d =2.25 l B CA d =0.75 l E F D d = l d =0.75 l d =1.25 l d =1.75 l d =2.25 l FIG. 4. (Color online) (a) Schematic illustration of the large triangular supercells with two types of PhCs. The outer PhC hasthe displacement d , while the inner PhC has the displacement 3 l − d . Several cases with different d ’s are shown in (a). (b)Eigen-frequencies of the photons as functions of the geometry parameter d . The green-gray regions represent the bulk states,the green regions represent the edge states, while the purple and blue curves represent the type-I and type-II corner states,respectively. (c) Electric field patterns of corner states and edge states with different d . Throughout this paper, the electricfield patterns of the corner states are given by the superpostion of | E z | of the three degenerate corner states. In the calculation,the side length of the supercell is 10 a , while the inside structure has a side length of 4 a . of “charge” through the local density of states (LDOS), ρ e ( r , E ). In electronic systems, the charge contributedby the filling of the valence bands in the j -th unit-cell isgiven by Q j,e = e (cid:90) E gap dE (cid:90) j d r ρ e ( r , E ) . (3)Here, e is the charge of an electron, E gap is an energyin the topological band gap which is below the eigen-energies of the edge and corner states. In the above equa-tion, the integration over the position r is defined withinthe j -th unit-cell. The filling of all the valence bandsbelow the topological band gap contributes a fractionalcharge in the corner region. It was predicted that thefractional corner charge eQ c is completely determined bythe topological indices of the bulk bands as follows, Q c = −
13 ([ K ] + [ K ]) mod 1 . (4)The actual size of the corner region depends on the spe-cific model. However, one can often choose one or a few unit-cells around the corner boundary to converge thefractional corner charge.We then check the theoretical prediction in photonicsystem by calculating the following quantity which is theanalog of the “charge” in the j -th unit-cell in the photonicsystem, Q j = (cid:90) f gap df (cid:90) j d r ρ p ( r , f ) . (5)Here, we omitted the elementary charge e which does nothave a physical meaning in photonics. The integrationover frequency is from 0 to a frequency in the band gap f gap which is below the eigen-frequencies of the edge andcorner states. The photonic LDOS is calculated throughthe following spectral decomposition of all the photoniceigenstates of the valence bulk bands ρ p ( r , f ) = (cid:88) n Γ π (cid:15) ( r ) | E ( n ) z ( r ) | ( f − f n ) + Γ . (6)Here, n labels the photonic eigenstates of the valencebulk bands, and Γ → CornerEdge
Bulk
Bulk F r e qu e n c y ( c / a ) (a) F r e qu e n c y ( c / a ) F r e qu e n c y ( c / a ) Solution number15 F r e qu e n c y ( c / a ) Bulk CornerEdgeBulk(b)(c) (d)
30 450.40.30.20.1 0.50.40.30.20.10.50.40.30.20.1 00.50.40.30.20.1 Solution number15 30 450Solution number15 30 450 Solution number15 30 450 d= ld= l d= l d= l FIG. 5. Fractional “charges” in the triangular supercell withperfect electric conductor boundary conditions. Only thecharges of the corner unit-cells and the bulk unit-cells areshown in the figure, as indicated by the blue areas. Fourcases are considered (a) d = 0 . l , (b) d = 0 . l , (c) d = 2 . l and (d) d = 2 . l . converges the calculation. E ( n ) z ( r ) is the scaled electricfield distribution of the n -th photonic eigenstate whichsatisfies the following normalization condition1 = (cid:90) d r (cid:15) ( r ) | E ( n ) z ( r ) | (7)where the integral of r is over the whole photonic systemand (cid:15) ( r ) is the position-dependent relative permittivity.The photonic “charge” defined above does have a phys-ical meaning. It represents the number of the photonicmodes contributed from the j -th unit-cell from the va-lence bulk bands. We calculate the photonic “charge”for each unit-cell and present the results in Fig. 5 forvarious configurations.For all the four cases considered in Fig. 5, the calcu-lated charge for the bulk unit-cells are close to 1. This isconsistent with the fact that there is only one band be- low the band gap, i.e., each unit-cell contributes a singlecharge (mode) to the bulk band. Figs. 5(a) and 5(c) showthat for both d = 0 . l and d = 2 . l (i.e., χ = [0 , d = 0 . l (i.e., χ = [ − , Q c = 0 which is indicated by that the corner unit-cellhas a charge very close to 0. In this case, despite that theband gap carries higher-order topology and the resultantcorner states, the corner charge vanishes, which is con-sistent with the theoretical prediction given in Eq. (4).Fig. 5(d) shows that for d = 2 . l (i.e., χ = [ − , Q c = 1 / /
3. These photonic charges can be measured throughthe classical or quantum versions of the Purcell effect, asindicated by Ref. 45.
V. CONCLUSION
In conclusion, we demonstrate that rich higher-ordertopological phases and multiple phase transitions can beobtained in C symmetric PhCs by tuning a single geom-etry parameter d . These higher-order topological phasesyield intriguing multidimensional topological phenomenawhere the corner and edge states can be tuned in versatileways. Our study shows that continuously configurabledielectric PhCs can be useful in generating topologicalphotonic circuits with tunable edge and corner states.The emergent fractional photonic charge indicates thatphotonic systems can be powerful in revealing the funda-mental properties of topological bands. ACKNOWLEDGMENTS
H.-X. Wang and L. Liang contributed equally to thiswork. This work is supported by the National NaturalScience Foundation of China under Grant No. 11904060,12074281. J.-H. Jiang is supported by the Jiangsuspecially-appointed professor funding, and a projectfunded by the Priority Academic Program Developmentof Jiangsu Higher Education Institutions (PAPD). ∗ [email protected] † [email protected] M. Z. Hasan and C. L. Kane, Colloquium: Topo-logical insulators, Rev. Mod. Phys. , 3045 (2010). https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.82.3045 X.-L. Qi and S.-C. Zhang, Topological insulatorsand superconductors, Rev. Mod. Phys. , 1057(2011). https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.83.1057 T. Ozawa, et al ., Topological photonics, Rev. Mod.Phys. , 015006 (2019). https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.91.015006 G. Ma, M. Xiao, and C. T. Chan, Topologicalphases in acoustic and mechanical systems, Nat. Rev.Phys. , 281 (2019). X. Zhang, M. Xiao, Y. Cheng, M.-H. Lu, and J.Christensen, Topological sound, Commun. Phys. , 97 (2018). W. A. Benalcazar, B. A. Bernevig, and T. L.Hughes, Quantized electric multipole insulators, Sci-ence , 61 (2017). https://science.sciencemag.org/content/357/6346/61.abstract W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes,Phys. Rev. B , 245115 (2017). https://journals.aps.org/prb/abstract/10.1103/PhysRevB.96.245115 S. Mittal, V.V. Orre, G. Zhu, M. A. Gorlach, A. Poddubny,and M. Hafezi, Photonic quadrupole topological phases,Nat. Photon. , 692 (2019). M. Serra-Garcia, V. Peri, R. Susstrunk, O. R. Bilal,T. Larsen, L. G. Villanueva, and S. D. Huber, Ob-servation of a phononic quadrupole topological insula-tor, Nature , 342 (2018). C. W. Peterson, W. A. Benalcazar, T. L. Hughes, and G.Bahl, A quantized microwave quadrupole insulator withtopologically protected corner states, Nature , 346(2018). S. Imhof, et al ., Topolectrical-circuit realization of topo-logical corner modes, Nat. Phys. , 925 (2018). S. Franca, J. van den Brink, and I. C. Fulga, Ananomalous higher-order topological insulator, Phys. Rev.B , 201114(R) (2018). https://journals.aps.org/prb/abstract/10.1103/PhysRevB.98.201114 X. Zhang, Z.-K. Lin, H.-X. Wang, Z. Xiong, Y. Tian,M.-H. Lu, Y.-F. Chen, and J.-H. Jiang, Symmetry-protected hierarchy of anomalous multipole topologicalband gaps in nonsymmorphic metacrystals, Nat. Com-mun. , 65 (2020). L. He, Z. Addison, E.J. Mele, and B. Zhen, Quadrupoletopological photonic crystals, Nat. Commun. ,3119 (2020). Y. Qi, C. Qiu, M. Xiao, H. He, M. Ke, and Z. Liu, AcousticRealization of Quadrupole Topological Insulators, Phys.Rev. Lett. , 206601 (2020). https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.124.206601 Y. Chen, Z.-K. Lin, H. Chen, and J.-H. Jiang, Plasmon-polaritonic quadrupole topological insulators, Phys. Rev. B , 041109(R) (2020). https://journals.aps.org/prb/abstract/10.1103/PhysRevB.101.041109 J. Langbehn, Y. Peng, L. Trifunovic, F. v. Oppen,and P. W. Brouwer, Reflection-Symmetric Second-OrderTopological Insulators and Superconductors, Phys. Rev.Lett. , 246401 (2017). https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.119.246401 Z. Song, Z. Fang, and C. Fang, ( d − , 246402(2017). https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.119.246402 F. Schindler, A. M. Cook, M. G. Vergniory, Z.Wang, S. S. P. Parkin, B. A. Bernevig, and T. Ne-upert, Higher-order topological insulators, Sci. Adv. , eaat0346 (2018). https://advances.sciencemag.org/content/4/6/eaat0346.abstract F. Schindler, Z. J. Wang, M. G. Vergniory, A. M. Cook,A. Murani, S. Sengupta, A. Y. Kasumov, R. Deblock, S.Jeon, I. Drozdov, H. Bouchiat, S. Gueron, A. Yazdani,B. A. Bernevig, and T. Neupert, Higher-order topology in bismuth, Nat. Phys. , 918 (2018). M. Ezawa, Higher-Order Topological Insulators andSemimetals on the Breathing Kagome and Py-rochlore Lattices, Phys. Rev. Lett. , 026801(2018). https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.120.026801 A. E. Hassan, F. K. Kunst, A. Moritz, G. Andler, E. J.Bergholtz, and M. Bourennane, Corner states of light inphotonic waveguides, Nat. Photon. , 697 (2019). H. Xue, Y. Yang, F. Gao, Y. Chong, and B. Zhang,Acoustic higher-order topological insulator on a kagomelattice, Nat. Mater. , 108 (2019). X. Ni, M.Weiner, A. Alu, and A. B. Khanikaev, Ob-servation of higher-order topological acoustic statesprotected by generalized chiral symmetry, Nat. Mater. , 113 (2019). B.-Y. Xie, H.-F. Wang, H.-X. Wang, X.-Y. Zhu, J.-H.Jiang, M.-H. Lu, and Y.-F. Chen, Second-order pho-tonic topological insulator with corner states, Phys. Rev.B , 205147 (2018). https://journals.aps.org/prb/abstract/10.1103/PhysRevB.98.205147 X. Zhang, H.-X. Wang, Z.-K. Lin, Y. Tian, B. Xie, M.-H.Lu, Y.-F. Chen, and J.-H. Jiang, Second-order topologyand multidimensional topological transitions in sonic crys-tals, Nat. Phys. , 582 (2019). S. Liu, W. Gao, Q. Zhang, S. Ma, L. Zhang, C. Liu, Y. J.Xiang, T. J. Cui, and S. Zhang, Topologically ProtectedEdge State in Two-Dimensional Su–Schrieffer–HeegerCircuit, Research , 8609875 (2019). https://spj.sciencemag.org/research/2019/8609875 Y. Ota, F. Liu, R. Katsumi, K. Watanabe, K. Wak-abayashi,Y. Arakawa, and S. Iwamoto, Photonic crystalnanocavity based on a topological corner state, Optica , 786 (2019). Z. Zhang, H. Long, C. Liu, C. Shao, Y. Cheng, X. Liu,and J. Christensen, Deep-Subwavelength Holey Acous-tic Second-Order Topological Insulators, Adv. Mater. ,1904682 (2019). https://onlinelibrary.wiley.com/doi/full/10.1002/adma.201904682 H. Fan, B. Xia, L. Tong, S. Zheng, and D.Yu, Elastic Higher-Order Topological Insulator withTopologically Protected Corner States, Phys. Rev.Lett. , 204301 (2019). https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.122.204301 X.-D. Chen, W.-M. Deng, F.-L. Shi, F.-L. Zhao, M. Chen,and J.-W. Dong, Direct Observation of Corner States inSecond-Order Topological Photonic Crystal Slabs, Phys.Rev. Lett. , 233902 (2019). https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.122.233902 B.-Y. Xie, G.-X. Su, H.-F. Wang, H. Su, X.-P. Shen,P. Zhan, M.-H. Lu, Z.-L. Wang, and Y.-F. Chen, Visu-alization of Higher-Order Topological Insulating Phasesin Two-Dimensional Dielectric Photonic Crystals, Phys.Rev. Lett. , 233903 (2019). https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.122.233903 L. Zhang, Y. Yang, P. Qin, Q. Chen, F. Gao, E. Li, J.-H.Jiang, B. Zhang, and H. Chen, Higher-Order Topologi-cal States in Surface-Wave Photonic Crystals, Adv. Sci. , https://onlinelibrary.wiley.com/doi/full/10.1002/advs.201902724 B. J. Wieder, Z. Wang, J. Cano, X. Dai, L. M. Schoop,B. Bradlyn, and B. A. Bernevig, Strong and fragile topo-logical Dirac semimetals with higher-order Fermi arcs,Nat. Commun. , 627 (2020). Q.-B. Zeng, Y.-B. Yang, and Y. Xu, Higher-order topo-logical insulators and semimetals in generalized Aubry-Andr´e-Harper models, Phys. Rev. B , 241104(R)(2020). https://journals.aps.org/prb/abstract/10.1103/PhysRevB.101.241104 X. Ni, M. Li, M.Weiner, A. Alu, and A. B. Khanikaev,Demonstration of a quantized acoustic octupole topolog-ical insulator, Nat. Commun. , 2108 (2020). H. Xue, Y. Ge, H.-X. Sun, Q. Wang, D. Jia, Y.-J. Guan, S.-Q. Yuan, Y. Chong, and B. Zhang, Ob-servation of an acoustic octupole topological insulator,Nat. Commun. , 2442 (2020). J. Noh, W. A. Benalcazar, S. Huang, M. J. Collins, K. P.Chen, T. L. Hughes, and M. C. Rechtsman, Topologicalprotection of photonic mid-gap defect modes, Nat. Pho-ton. , 408 (2018). H. Xue, Y. Yang, G. Liu, F. Gao, Y. Chong, andB. Zhang, Realization of an Acoustic Third-OrderTopological Insulator, Phys. Rev. Lett. , 244301(2019). https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.122.244301 M. Weiner, X. Ni, M. Y. Li, A. Alu, and A. B. Khanikaev,Demonstration of a 3rd order hierarchy of higher order topological states in a three-dimensional acoustic meta-material, Sci. Adv. , eaay4166 (2020). https://advances.sciencemag.org/content/6/13/eaay4166 M. Li, D. Zhirihin, M. Gorlach, X. Ni, D. Filonov, A.Slobozhanyuk, A. Alu, and A. B. Khanikaev, Higher-ordertopological states in photonic kagome crystals with long-range interactions, Nat. Photonics , 89-94 (2020). Y. Chen, X. Lu, and H. Chen, Effect of truncationon photonic corner states in a Kagome lattice, Opt.Lett. W. A. Beanalcazar, T. Li, and T. L. Hughes, Quan-tization of fractional corner charge in C n -symmetrichigher-order topological crystalline insulators, Phys. Rev.B , 245151 (2019). https://journals.aps.org/prb/abstract/10.1103/PhysRevB.99.245151 Z. Xiong, Z.-K. Lin, H.-X. Wang, X. Zhang, M.-H. Lu,Y.-F. Chen, and J.-H. Jiang, Corner states and topo-logical transitions in two-dimensional higher-order topo-logical sonic crystals wit inversion symmetry, Phy. Rev.B , 125144 (2020). https://journals.aps.org/prb/abstract/10.1103/PhysRevB.102.125144 Y. Liu, S. Leung, F.-F. Li, Z.-K. Lin, X. Tao, Y. Poo, andJ.-H. Jiang, Experimental discovery of bulk-disclinationcorrespondence. Nature 589, 381-385 (2021). X. Zhu, H.-X. Wang, C. Xu, J.-H. Jiang, and S. John,Topological transitions in continuously deformed photoniccrystals, Phy. Rev. B , 085148 (2018)., 085148 (2018).