Hidden multi-topological phases mediated by constrained inter-cell coupling

Ziteng Wang; Domenico Bongiovanni; Xiangdong Wang; Zhichan Hu; Dario Jukić; Daohong Song; Jingjun Xu; Roberto Morandotti; Zhigang Chen; Hrvoje Buljan

doi:10.1186/s43593-025-00118-5

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Hidden multi-topological phases mediated by constrained inter-cell coupling

eLight Vol. 6, (2026)

Research Article | Updated：2026-03-16

- Hidden multi-topological phases mediated by constrained inter-cell coupling
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- DOI：10.1186/s43593-025-00118-5
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- Received：11 July 2025，
  
  Revised：2025-10-30，
  
  Accepted：12 November 2025，
  
  Online First：13 January 2026，
  
  Published：2026-12
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Ziteng Wang, Domenico Bongiovanni, Xiangdong Wang, et al. Hidden multi-topological phases mediated by constrained inter-cell coupling[J]. eLight, 2026, 6.
DOI：

Ziteng Wang, Domenico Bongiovanni, Xiangdong Wang, et al. Hidden multi-topological phases mediated by constrained inter-cell coupling[J]. eLight, 2026, 6. DOI： 10.1186/s43593-025-00118-5.

摘要

Abstract

The discovery of topological phases of matter and topological boundary states had a tremendous impact on condensed matter physics

photonics

and material sciences

where topological phases are defined via energy bands

described by the topological band theory. However

there are topological materials that cannot be described by this theory

which support non-trivial boundary states but are little-known and largely unexplored. Here

we uncover a new class of topological phases—termed "multi-topological phase" (MTPs)—arising from constrained inter-cell coupling in lattice systems

and experimentally demonstrate them in a photonic platform. The MTP features multiple sets of boundary states

where each set is associated with one distinct topological invariant. Unlike conventional topological phases

the MTP cannot be identified via the original band structure

being a "hidden" topological phase

where the phase transition can occur without band-gap closing. We present typical examples of MTPs in both one- and two-dimensional structures

as well as in indirectly gapped Chern insulators

beyond the regime where the conventional bulk-boundary correspondence predicts the existence of boundary states. Furthermore

we directly observe the MTPs in the first two examples using laser-written photonic lattices. Our work offers a new design strategy for topological materials

paving the way for future exploration and applications in photonics.

关键词

Keywords

references

M.Z. Hasan , C.L. Kane , Colloquium: topological insulators . Rev. Mod. Phys. 82 , 3045 - 3067 ( 2010 ). http://doi.org/10.1103/RevModPhys.82.3045 http://doi.org/10.1103/RevModPhys.82.3045

X.-L. Qi , S.-C. Zhang , Topological insulators and superconductors . Rev. Mod. Phys. 83 , 1057 - 1110 ( 2011 ). http://doi.org/10.1103/RevModPhys.83.1057 http://doi.org/10.1103/RevModPhys.83.1057

C.-K. Chiu , J.C.Y. Teo , A.P. Schnyder , S. Ryu , Classification of topological quantum matter with symmetries . Rev. Mod. Phys. 88 ,( 2016 ). http://doi.org/10.1103/RevModPhys.88.035005 http://doi.org/10.1103/RevModPhys.88.035005

D.J. Thouless , M. Kohmoto , M.P. Nightingale , M. den Nijs , Quantized hall conductance in a two-dimensional periodic potential . Phys. Rev. Lett. 49 , 405 - 408 ( 1982 ). http://doi.org/10.1103/PhysRevLett.49.405 http://doi.org/10.1103/PhysRevLett.49.405

F.D.M. Haldane , Model for a quantum hall effect without Landau levels: condensed-matter realization of the "parity anomaly" . Phys. Rev. Lett. 61 , 2015 - 2018 ( 1988 ). http://doi.org/10.1103/PhysRevLett.61.2015 http://doi.org/10.1103/PhysRevLett.61.2015

C.L. Kane , E.J. Mele , Z 2 topological order and the quantum spin Hall effect . Phys. Rev. Lett. 95 ,( 2005 ). http://doi.org/10.1103/PhysRevLett.95.146802 http://doi.org/10.1103/PhysRevLett.95.146802

M. König et al. , Quantum spin hall insulator state in HgTe quantum wells . Science 318 , 766 - 770 ( 2007 ). http://doi.org/10.1126/science.1148047 http://doi.org/10.1126/science.1148047

C.-Z. Chang et al. , Experimental observation of the quantum anomalous hall effect in a magnetic topological insulator . Science 340 , 167 - 170 ( 2013 ). http://doi.org/10.1126/science.1234414 http://doi.org/10.1126/science.1234414

L. Lu , J.D. Joannopoulos , M. Soljačić , Topological photonics . Nat. Photon. 8 ( 11 ), 821 - 829 ( 2014 ). http://doi.org/10.1038/nphoton.2014.248 http://doi.org/10.1038/nphoton.2014.248

T. Ozawa et al. , Topological photonics . Rev. Mod. Phys. 91 ,( 2019 ). http://doi.org/10.1103/RevModPhys.91.015006 http://doi.org/10.1103/RevModPhys.91.015006

G. Ma , M. Xiao , C.T. Chan , Topological phases in acoustic and mechanical systems . Nat. Rev. Phys. 1 , 281 - 294 ( 2019 ). http://doi.org/10.1038/s42254-019-0030-x http://doi.org/10.1038/s42254-019-0030-x

H. Xue , Y. Yang , B. Zhang , Topological acoustics . Nat. Rev. Mater. 7 ( 12 ), 974 - 990 ( 2022 ). http://doi.org/10.1038/s41578-022-00465-6 http://doi.org/10.1038/s41578-022-00465-6

J. Kruthoff , J. de Boer , J. van Wezel , C.L. Kane , R.-J. Slager , Topological classification of crystalline insulators through band structure combinatorics . Phys. Rev. X 7 ( 4 ),( 2017 ). http://doi.org/10.1103/PhysRevX.7.041069 http://doi.org/10.1103/PhysRevX.7.041069

B. Bradlyn et al. , Topological quantum chemistry . Nature 547 , 298 - 305 ( 2017 ). http://doi.org/10.1038/nature23268 http://doi.org/10.1038/nature23268

Z. Wang , Y. Chong , J.D. Joannopoulos , M. Soljačić , Observation of unidirectional backscattering-immune topological electromagnetic states . Nature 461 , 772 - 775 ( 2009 ). http://doi.org/10.1038/nature08293 http://doi.org/10.1038/nature08293

M.C. Rechtsman et al. , Photonic floquet topological insulators . Nature 496 , 196 - 200 ( 2013 ). http://doi.org/10.1038/nature12066 http://doi.org/10.1038/nature12066

M. Hafezi , S. Mittal , J. Fan , A. Migdall , J.M. Taylor , Imaging topological edge states in silicon photonics . Nat. Photon. 7 , 1001 - 1005 ( 2013 ). http://doi.org/10.1038/nphoton.2013.274 http://doi.org/10.1038/nphoton.2013.274

G. Jotzu et al. , Experimental realization of the topological Haldane model with ultracold fermions . Nature 515 , 237 - 240 ( 2014 ). http://doi.org/10.1038/nature13915 http://doi.org/10.1038/nature13915

C. He et al. , Acoustic topological insulator and robust one-way sound transport . Nat. Phys. 12 , 1124 - 1129 ( 2016 ). http://doi.org/10.1038/nphys3867 http://doi.org/10.1038/nphys3867

M.A. Bandres et al. , Topological insulator laser: experiments . Science 359 ,( 2018 ). http://doi.org/10.1126/science.aar4005 http://doi.org/10.1126/science.aar4005

Y. Zeng et al. , Electrically pumped topological laser with valley edge modes . Nature 578 , 246 - 250 ( 2020 ). http://doi.org/10.1038/s41586-020-1981-x http://doi.org/10.1038/s41586-020-1981-x

A. Dikopoltsev et al. , Topological insulator vertical-cavity laser array . Science 373 , 1514 - 1517 ( 2021 ). http://doi.org/10.1126/science.abj2232 http://doi.org/10.1126/science.abj2232

L. Yang , G. Li , X. Gao , L. Lu , Topological-cavity surface-emitting laser . Nat. Photon. 16 , 279 - 283 ( 2022 ). http://doi.org/10.1038/s41566-022-00972-6 http://doi.org/10.1038/s41566-022-00972-6

C.L. Kane , T.C. Lubensky , Topological boundary modes in isostatic lattices . Nat. Phys. 10 , 39 - 45 ( 2014 ). http://doi.org/10.1038/nphys2835 http://doi.org/10.1038/nphys2835

Z.-Q. Jiao et al. , Experimentally detecting quantized Zak phases without chiral symmetry in photonic lattices . Phys. Rev. Lett. 127 ,( 2021 ). http://doi.org/10.1103/PhysRevLett.127.147401 http://doi.org/10.1103/PhysRevLett.127.147401

Z. Wang et al. , Sub-symmetry-protected topological states . Nat. Phys. 19 , 992 - 998 ( 2023 ). http://doi.org/10.1038/s41567-023-02011-9 http://doi.org/10.1038/s41567-023-02011-9

S. Verma , M.J. Park , Non-bloch band theory of subsymmetry-protected topological phases . Phys. Rev. B 110 ( 3 ),( 2024 ). http://doi.org/10.1103/PhysRevB.110.035424 http://doi.org/10.1103/PhysRevB.110.035424

W.A. Benalcazar , B.A. Bernevig , T.L. Hughes , Quantized electric multipole insulators . Science 357 , 61 - 66 ( 2017 ). http://doi.org/10.1126/science.aah6442 http://doi.org/10.1126/science.aah6442

C.W. Peterson , W.A. Benalcazar , T.L. Hughes , G. Bahl , A quantized microwave quadrupole insulator with topologically protected corner states . Nature 555 , 346 - 350 ( 2018 ). http://doi.org/10.1038/nature25777 http://doi.org/10.1038/nature25777

L. Fu , Topological crystalline insulators . Phys. Rev. Lett. 106 ,( 2011 ). http://doi.org/10.1103/PhysRevLett.106.106802 http://doi.org/10.1103/PhysRevLett.106.106802

H.C. Po , H. Watanabe , A. Vishwanath , Fragile topology and Wannier obstructions . Phys. Rev. Lett. 121 ( 12 ),( 2018 ). http://doi.org/10.1103/PhysRevLett.121.126402 http://doi.org/10.1103/PhysRevLett.121.126402

A. Bouhon , A.M. Black-Schaffer , R.-J. Slager , Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time-reversal symmetry . Phys. Rev. B 100 ( 19 ),( 2019 ). http://doi.org/10.1103/PhysRevB.100.195135 http://doi.org/10.1103/PhysRevB.100.195135

Y. Hwang , J. Ahn , B.-J. Yang , Fragile topology protected by inversion symmetry: diagnosis, bulk-boundary correspondence, and Wilson loop . Phys. Rev. B 100 ,( 2019 ). http://doi.org/10.1103/PhysRevB.100.205126 http://doi.org/10.1103/PhysRevB.100.205126

A. Bouhon , T. Bzdušek , R.-J. Slager , Geometric approach to fragile topology beyond symmetry indicators . Phys. Rev. B 102 ( 11 ),( 2020 ). http://doi.org/10.1103/PhysRevB.102.115135 http://doi.org/10.1103/PhysRevB.102.115135

V. Peri et al. , Experimental characterization of fragile topology in an acoustic metamaterial . Science 367 , 797 - 800 ( 2020 ). http://doi.org/10.1126/science.aaz7654 http://doi.org/10.1126/science.aaz7654

Z.-D. Song , L. Elcoro , B.A. Bernevig , Twisted bulk-boundary correspondence of fragile topology . Science 367 , 794 - 797 ( 2020 ). http://doi.org/10.1126/science.aaz7650 http://doi.org/10.1126/science.aaz7650

A. Nelson , T. Neupert , T. Bzdušek , A. Alexandradinata , Multicellularity of delicate topological insulators . Phys. Rev. Lett. 126 ,( 2021 ). http://doi.org/10.1103/PhysRevLett.126.216404 http://doi.org/10.1103/PhysRevLett.126.216404

A. Nelson , T. Neupert , A. Alexandradinata , T. Bzdušek , Delicate topology protected by rotation symmetry: crystalline Hopf insulators and beyond . Phys. Rev. B 106 ( 7 ),( 2022 ). http://doi.org/10.1103/PhysRevB.106.075124 http://doi.org/10.1103/PhysRevB.106.075124

Y. Wang et al. , Hybrid topological photonic crystals . Nat. Commun. 14 , 4457 ( 2023 ). http://doi.org/10.1038/s41467-023-40172-6 http://doi.org/10.1038/s41467-023-40172-6

M.S. Hossain et al. , A hybrid topological quantum state in an elemental solid . Nature 628 , 527 - 533 ( 2024 ). http://doi.org/10.1038/s41586-024-07203-8 http://doi.org/10.1038/s41586-024-07203-8

J. Zak , Berry's phase for energy bands in solids . Phys. Rev. Lett. 62 , 2747 - 2750 ( 1989 ). http://doi.org/10.1103/PhysRevLett.62.2747 http://doi.org/10.1103/PhysRevLett.62.2747

C. Fang , M.J. Gilbert , B.A. Bernevig , Bulk topological invariants in noninteracting point group symmetric insulators . Phys. Rev. B 86 ( 11 ),( 2012 ). http://doi.org/10.1103/PhysRevB.86.115112 http://doi.org/10.1103/PhysRevB.86.115112

W.A. Benalcazar , T. Li , T.L. Hughes , Quantization of fractional corner charge in C n -symmetric higher-order topological crystalline insulators . Phys. Rev. B 99 ( 24 ),( 2019 ). http://doi.org/10.1103/PhysRevB.99.245151 http://doi.org/10.1103/PhysRevB.99.245151

C.W. Peterson , T. Li , W.A. Benalcazar , T.L. Hughes , G. Bahl , A fractional corner anomaly reveals higher-order topology . Science 368 , 1114 - 1118 ( 2020 ). http://doi.org/10.1126/science.aba7604 http://doi.org/10.1126/science.aba7604

S. Xia et al. , Unconventional flatband line states in photonic Lieb lattices . Phys. Rev. Lett. 121 ,( 2018 ). http://doi.org/10.1103/PhysRevLett.121.263902 http://doi.org/10.1103/PhysRevLett.121.263902

Z. Hu et al. , Nonlinear control of photonic higher-order topological bound states in the continuum . Light Sci. Appl. 10 , 164 ( 2021 ). http://doi.org/10.1038/s41377-021-00607-5 http://doi.org/10.1038/s41377-021-00607-5

W.A. Benalcazar , A. Cerjan , Chiral-symmetric higher-order topological phases of matter . Phys. Rev. Lett. 128 ,( 2022 ). http://doi.org/10.1103/PhysRevLett.128.127601 http://doi.org/10.1103/PhysRevLett.128.127601

Z. Hu et al. , Topological orbital angular momentum extraction and twofold protection of vortex transport . Nat. Photon. 19 , 162 - 169 ( 2025 ). http://doi.org/10.1038/s41566-024-01564-2 http://doi.org/10.1038/s41566-024-01564-2

W.P. Su , J.R. Schrieffer , A.J. Heeger , Solitons in polyacetylene . Phys. Rev. Lett. 42 , 1698 - 1701 ( 1979 ). http://doi.org/10.1103/PhysRevLett.42.1698 http://doi.org/10.1103/PhysRevLett.42.1698

Z. Yang , E. Lustig , Y. Lumer , M. Segev , Photonic Floquet topological insulators in a fractal lattice . Light Sci. Appl. 9 , 128 ( 2020 ). http://doi.org/10.1038/s41377-020-00354-z http://doi.org/10.1038/s41377-020-00354-z

T. Biesenthal et al. , Fractal photonic topological insulators . Science 376 , 1114 - 1119 ( 2022 ). http://doi.org/10.1126/science.abm2842 http://doi.org/10.1126/science.abm2842

N.P. Mitchell , L.M. Nash , D. Hexner , A.M. Turner , W.T.M. Irvine , Amorphous topological insulators constructed from random point sets . Nat. Phys. 14 , 380 - 385 ( 2018 ). http://doi.org/10.1038/s41567-017-0024-5 http://doi.org/10.1038/s41567-017-0024-5

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