- Physical Review Journals
All JournalsPhysics Magazine

Physical Review X

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Observation of Anomalous Floquet Non-Abelian Topological Insulators

Huahui Qiu*, Shuaishuai Tong*, Qicheng Zhang, Kun Zhang, and Chunyin Qiu†

  • Key Laboratory of Artificial Micro- and Nano-Structures of Ministry of Education and School of Physics and Technology, Wuhan University, Wuhan 430072, China

  • *These authors contributed equally to this work.
  • †Contact author: cyqiu@whu.edu.cn
PDF

Phys. Rev. X 16, 011061 – Published 20 March, 2026

DOI: https://doi.org/10.1103/qn87-bm33

Abstract

Non-Abelian topological phases, which go beyond traditional Abelian topological band theory, are garnering increasing attention. This is further spurred by periodic driving, leading to predictions of many novel multigap Floquet topological phases, including anomalous Euler and Dirac string phases induced by non-Abelian Floquet braiding, as well as Floquet non-Abelian topological insulators (FNTIs) that exhibit multifold bulk-edge correspondence. Here, we report the first experimental realization of anomalous FNTIs, which demonstrate topological edge modes in all three gaps despite having a trivial bulk charge. Concretely, we construct an experimentally feasible one-dimensional three-band Floquet model and implement it in acoustics by integrating time-periodic coupling circuits to static acoustic crystals. Furthermore, we observe counterintuitive topological interface modes in the domain wall formed by an anomalous FNTI and its counterpart with swapped driving sequences—modes previously inaccessible in Floquet Abelian systems. Our work paves the way for further experimental exploration of the uncharted nonequilibrium topological physics.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Topological materials are traditionally characterized by static bulk invariants that dictate the presence of protected edge states. We experimentally demonstrate that periodic driving in a non-Abelian system breaks this paradigm by realizing a Floquet topological phase governed by non-Abelian rules where the multiplication order of topological charges is noncommutative. We observe that, despite having a trivial bulk quaternion charge, this anomalous Floquet non-Abelian topological insulator exhibits robust edge states across all band gaps, including the characteristic π gap of Floquet systems. Moreover, we detect topological interface modes at the boundary between two such insulators with swapped driving sequences, a signature of noncommutative dynamics inaccessible in Abelian systems. Our work establishes an experimental framework for realizing and probing this phase, enabling further exploration of the intersection between periodic driving, topology, and non-Abelian physics.

Article Text

Supplemental Material

References (62)

  1. B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science 314, 1757 (2006).
  2. M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Quantum spin Hall insulator state in HgTe quantum wells, Science 318, 766 (2007).
  3. M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
  4. J. E. Moore, The birth of topological insulators, Nature (London) 464, 194 (2010).
  5. X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
  6. Q. Wu, A. A. Soluyanov, and T. Bzdusek, Non-Abelian band topology in noninteracting metals, Science 365, 1273 (2019).
  7. J. Ahn, S. Park, and B.-J. Yang, Failure of Nielsen-Ninomiya Theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: Application to twisted bilayer graphene at magic angle, Phys. Rev. X 9, 021013 (2019).
  8. A. Tiwari and T. Bzdušek, Non-Abelian topology of nodal-line rings in PT-symmetric systems, Phys. Rev. B 101, 195130 (2020).
  9. A. Bouhon, Q. Wu, R.-J. Slager, H. Weng, O. V. Yazyev, and T. Bzdušek, Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe, Nat. Phys. 16, 1137 (2020).
  10. Y. Yang, B. Yang, G. Ma, J. Li, S. Zhang, and C. T. Chan, Non-Abelian physics in light and sound, Science 383, eadf9621 (2024).
  11. E. Yang, B. Yang, O. You, H. C. Chan, P. Mao, Q. Guo, S. Ma, L. Xia, D. Fan, Y. Xiang, and S. Zhang, Observation of non-Abelian nodal links in photonics, Phys. Rev. Lett. 125, 033901 (2020).
  12. D. Wang, B. Yang, M. Wang, R. Y. Zhang, X. Li, Z. Q. Zhang, S. Zhang, and C. T. Chan, Observation of non-Abelian charged nodes linking nonadjacent gaps, Phys. Rev. Lett. 129, 263604 (2022).
  13. D. Wang, Y. Wu, Z. Q. Zhang, and C. T. Chan, Non-Abelian frame charge flow in photonic media, Phys. Rev. X 13, 021024 (2023).
  14. Y. Hu, M. Tong, T. Jiang, J. H. Jiang, H. Chen, and Y. Yang, Observation of two-dimensional time-reversal broken non-Abelian topological states, Nat. Commun. 15, 10036 (2024).
  15. B. Jiang, A. Bouhon, Z.-K. Lin, X. Zhou, B. Hou, F. Li, R.-J. Slager, and J.-H. Jiang, Experimental observation of non-Abelian topological acoustic semimetals and their phase transitions, Nat. Phys. 17, 1239 (2021).
  16. M. Wang, S. Liu, Q. Ma, R.-Y. Zhang, D. Wang, Q. Guo, B. Yang, M. Ke, Z. Liu, and C. T. Chan, Experimental observation of non-Abelian earring nodal links in phononic crystals, Phys. Rev. Lett. 128, 246601 (2022).
  17. H. Qiu, Q. Zhang, T. Liu, X. Fan, F. Zhang, and C. Qiu, Minimal non-Abelian nodal braiding in ideal metamaterials, Nat. Commun. 14, 1261 (2023).
  18. X.-C. Sun, J.-B. Wang, C. He, and Y.-F. Chen, Non-Abelian topological phases and their quotient relations in acoustic systems, Phys. Rev. Lett. 132, 216602 (2024).
  19. Q. Guo, T. Jiang, R. Y. Zhang, L. Zhang, Z. Q. Zhang, B. Yang, S. Zhang, and C. T. Chan, Experimental observation of non-Abelian topological charges and edge states, Nature (London) 594, 195 (2021).
  20. T. Jiang, Q. Guo, R. Y. Zhang, Z. Q. Zhang, B. Yang, and C. T. Chan, Four-band non-Abelian topological insulator and its experimental realization, Nat. Commun. 12, 6471 (2021).
  21. T. Kitagawa, E. Berg, M. Rudner, and E. Demler, Topological characterization of periodically driven quantum systems, Phys. Rev. B 82, 235114 (2010).
  22. N. H. Lindner, G. Refael, and V. Galitski, Floquet topological insulator in semiconductor quantum wells, Nat. Phys. 7, 490 (2011).
  23. R. Fleury, A. B. Khanikaev, and A. Alù, Floquet topological insulators for sound, Nat. Commun. 7, 11744 (2016).
  24. S. Yao, Z. Yan, and Z. Wang, Topological invariants of Floquet systems: General formulation, special properties, and Floquet topological defects, Phys. Rev. B 96, 195303 (2017).
  25. R. Roy and F. Harper, Periodic table for Floquet topological insulators, Phys. Rev. B 96, 155118 (2017).
  26. H. Hübener, M. A. Sentef, U. De Giovannini, A. F. Kemper, and A. Rubio, Creating stable Floquet–Weyl semimetals by laser-driving of 3D Dirac materials, Nat. Commun. 8, 1 (2017).
  27. A. Eckardt, Colloquium: Atomic quantum gases in periodically driven optical lattices, Rev. Mod. Phys. 89, 011004 (2017).
  28. M. S. Rudner and N. H. Lindner, Band structure engineering and non-equilibrium dynamics in Floquet topological insulators, Nat. Rev. Phys. 2, 229 (2020).
  29. C. Bao, P. Tang, D. Sun, and S. Zhou, Light-induced emergent phenomena in 2D materials and topological materials, Nat. Rev. Phys. 4, 33 (2021).
  30. M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems, Phys. Rev. X 3, 031005 (2013).
  31. R. J. Slager, A. Bouhon, and F. N. Unal, Non-Abelian Floquet braiding and anomalous Dirac string phase in periodically driven systems, Nat. Commun. 15, 1144 (2024).
  32. V. Karle, M. Lemeshko, A. Bouhon, R. J. Slager, and F. N. Unal, Anomalous multi-gap topological phases in periodically driven quantum rotors, Phys. Rev. A 113, 012216 (2026).
  33. T. Li and H. Hu, Floquet non-Abelian topological insulator and multifold bulk-edge correspondence, Nat. Commun. 14, 6418 (2023).
  34. F. Nathan and M. S. Rudner, Topological singularities and the general classification of Floquet–Bloch systems, New J. Phys. 17, 125014 (2015).
  35. M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Photonic Floquet topological insulators, Nature (London) 496, 196 (2013).
  36. Y.-G. Peng, C.-Z. Qin, D.-G. Zhao, Y.-X. Shen, X.-Y. Xu, M. Bao, H. Jia, and X.-F. Zhu, Experimental demonstration of anomalous Floquet topological insulator for sound, Nat. Commun. 7, 13368 (2016).
  37. L. J. Maczewsky, J. M. Zeuner, S. Nolte, and A. Szameit, Observation of photonic anomalous Floquet topological insulators, Nat. Commun. 8, 13756 (2017).
  38. S. Mukherjee, A. Spracklen, M. Valiente, E. Andersson, P. Ohberg, N. Goldman, and R. R. Thomson, Experimental observation of anomalous topological edge modes in a slowly driven photonic lattice, Nat. Commun. 8, 13918 (2017).
  39. S. Stutzer, Y. Plotnik, Y. Lumer, P. Titum, N. H. Lindner, M. Segev, M. C. Rechtsman, and A. Szameit, Photonic topological Anderson insulators, Nature (London) 560, 461 (2018).
  40. K. Wintersperger, C. Braun, F. N. Ünal, A. Eckardt, M. D. Liberto, N. Goldman, I. Bloch, and M. Aidelsburger, Realization of an anomalous Floquet topological system with ultracold atoms, Nat. Phys. 16, 1058 (2020).
  41. S. Mukherjee and M. C. Rechtsman, Observation of Floquet solitons in a topological bandgap, Science 368, 856 (2020).
  42. G. G. Pyrialakos, J. Beck, M. Heinrich, L. J. Maczewsky, N. V. Kantartzis, M. Khajavikhan, A. Szameit, and D. N. Christodoulides, Bimorphic Floquet topological insulators, Nat. Mater. 21, 634 (2022).
  43. Z. Cheng, R. W. Bomantara, H. Xue, W. Zhu, J. Gong, and B. Zhang, Observation of π/2 modes in an acoustic Floquet system, Phys. Rev. Lett. 129, 254301 (2022).
  44. W. Zhu, H. Xue, J. Gong, Y. Chong, and B. Zhang, Time-periodic corner states from Floquet higher-order topology, Nat. Commun. 13, 11 (2022).
  45. See Supplemental Material at http://link.aps.org/supplemental/10.1103/qn87-bm33 for further theoretical and experimental details, which includes Refs. [46–49].
  46. T. Li, J. Du, Q. Zhang, Y. Li, X. Fan, F. Zhang, and C. Qiu, Acoustic Möbius insulators from projective symmetry, Phys. Rev. Lett. 128, 116803 (2022).
  47. T. Li, L. Liu, Q. Zhang, and C. Qiu, Acoustic realization of projective mirror Chern insulators, Commun. Phys. 6, 268 (2023).
  48. Z. K. Lin, Y. Zhou, B. Jiang, B. Q. Wu, L. M. Chen, X. Y. Liu, L. W. Wang, P. Ye, and J. H. Jiang, Measuring entanglement entropy and its topological signature for phononic systems, Nat. Commun. 15, 1601 (2024).
  49. B. Jiang, A. Bouhon, S.-Q. Wu, Z.-L. Kong, Z.-K. Lin, R.-J. Slager, and J.-H. Jiang, Observation of an acoustic topological Euler insulator with meronic waves, Sci. Bull. 69, 1653 (2024).
  50. L. Zhang, Y. Yang, Y. Ge, Y. Guan, Q. Chen, Q. Yan, F. Chen, R. Xi, Y. Li, D. Jia, S. Yuan, H. Sun, H. Chen, and B. Zhang, Acoustic non-Hermitian skin effect from twisted winding topology, Nat. Commun. 12, 6297 (2021).
  51. Q. Zhang, Y. Li, H. Sun, X. Liu, L. Zhao, X. Feng, X. Fan, and C. Qiu, Observation of acoustic non-Hermitian bloch braids and associated topological phase transitions, Phys. Rev. Lett. 130, 017201 (2023).
  52. Z.-X. Chen, A. Chen, Y.-G. Peng, Z. W. Li, B. Liang, J. Yang, X.-F. Zhu, Y. Q. Lu, and J. C. Cheng, Observation of acoustic Floquet π modes in a time-varying lattice, Phys. Rev. B 109, L020302 (2024).
  53. S. Tong, Q. Zhang, L. Qi, G. Li, X. Feng, and C. Qiu, Observation of Floquet-bloch braids in non-Hermitian spatiotemporal lattices, Phys. Rev. Lett. 134, 126603 (2025).
  54. S. Tong, Q. Zhang, G. Li, K. Zhang, C. Xie, and C. Qiu, Observation of momentum-band topology in PT-symmetric Floquet lattices, Nat. Commun. 16, 9975 (2025).
  55. S. Tong, Q. Zhang, G. Li, K. Zhang, and C. Qiu, Acoustic realization of monoatomic topological space-time crystals, Newton 2, 2100304 (2026).
  56. T. T. Koutserimpas and R. Fleury, Nonreciprocal gain in non-Hermitian time-Floquet systems, Phys. Rev. Lett. 120, 087401 (2018).
  57. F. N. Ünal, A. Bouhon, and R.-J. Slager, Topological Euler class as a dynamical observable in optical lattices, Phys. Rev. Lett. 125, 053601 (2020).
  58. K. Zhang, Q. Zhang, S. Tong, W. Wu, X. Feng, and C. Qiu, Experimental observation of hidden multistability in nonlinear systems, Phys. Rev. Lett. 136, 037201 (2026).
  59. Z.-X. Chen, Y. Ru, G.-C. He, M.-H. Lu, Y.-F. Chen, Y.-Q. Lu, and Z.-G. Chen, Nonlinear coupling induced anomalous state transfer and complete multistate excitation via adiabatic control, Phys. Rev. Lett. 136, 037202 (2026).
  60. E. Lustig, Y. Sharabi, and M. Segev, Topological aspects of photonic time crystals, Optica 5, 1390 (2018).
  61. S. Franca, F. Hassler, and I. C. Fulga, Simulating Floquet topological phases in static systems, SciPost Phys. Core 4, 007 (2021).
  62. Q. Lin, T. Li, H. Hu, W. Yi, and P. Xue, Simulating Floquet non-Abelian topological insulator with photonic quantum walks arXiv:2508.06466.

Outline

Information

Phys. Rev. X 16, 011061– Published 20 March, 2026
  • Received 25 March 2025
  • Accepted 17 February 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation