Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).
Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).
Altland, A. & Zirnbauer, M. R. Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys. Rev. B 55, 1142 (1997).
Ryu, S., Schnyder, A. P., Furusaki, A. & Ludwig, A. W. Topological insulators and superconductors: tenfold way and dimensional hierarchy. New J. Phys. 12, 065010 (2010).
Schnyder, A. P., Ryu, S., Furusaki, A. & Ludwig, A. W. Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125 (2008).
Kitaev, A. Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22 (2009).
Chiu, C.-K., Teo, J. C., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016).
Hatsugai, Y. Chern number and edge states in the integer quantum Hall effect. Phys. Rev. Lett. 71, 3697 (1993).
Graf, G. M. & Porta, M. Bulk-edge correspondence for two-dimensional topological insulators. Commun. Math. Phys. 324, 851–895 (2013).
Yao, S. & Wang, Z. Edge states and topological invariants of non-Hermitian systems. Phys. Rev. Lett. 121, 086803 (2018).
Yang, Z., Zhang, K., Fang, C. & Hu, J. Non-Hermitian bulk-boundary correspondence and auxiliary generalized Brillouin zone theory. Phys. Rev. Lett. 125, 226402 (2020).
Longhi, S. Topological phase transition in non-Hermitian quasicrystals. Phys. Rev. Lett. 122, 237601 (2019).
Rudner, M. S. & Lindner, N. H. Band structure engineering and non-equilibrium dynamics in Floquet topological insulators. Nat. Rev. Phys. 2, 229–244 (2020).
Ahn, J., Park, S. & Yang, B.-J. 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).
Bouhon, A., Bzdusek, T. & Slager, R.-J. Geometric approach to fragile topology beyond symmetry indicators. Phys. Rev. B 102, 115135 (2020).
Zhao, W., Luo, X.-W., Zheng, W. & Duan, L.-M. Quantum simulation for topological Euler insulators. Commun. Phys. 5, 223 (2022).
Ünal, F. N., Bouhon, A. & Slager, R.-J. Topological Euler class as a dynamical observable in optical lattices. Phys. Rev. Lett. 125, 053601 (2020).
Wu, Q., Soluyanov, A. A. & Bzdušek, T. Non-Abelian band topology in noninteracting metals. Science 365, 1273–1277 (2019).
Bouhon, A. et al. Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe. Nat. Phys. 16, 1137–1143 (2020).
Qiu, H. et al. Minimal non-Abelian nodal braiding in ideal metamaterials. Nat. Commun. 14, 1261 (2023).
Tiwari, A. & Bzdušek, T. Non-Abelian topology of nodal-line rings in PT-symmetric systems. Phys. Rev. B 101, 195130 (2020).
Jiang, B. et al. Experimental observation of non-Abelian topological acoustic semimetals and their phase transitions. Nat. Phys. 17, 1239–1246 (2021).
Jiang, B. et al. Experimental observation of meronic topological acoustic Euler insulators. Sci. Bull. 69, 1653 (2024).
Hu, Y. et al. Observation of non-Abelian band topology without time-reversal symmetry. Nat. Commun. 15, 10036 (2024).
Jankowski, W. J. et al. Non-Abelian Hopf–Euler insulators. Phys. Rev. B 110, 075135 (2024).
Slager, R.-J., Bouhon, A. & Ünal, F. N. Non-Abelian Floquet braiding and anomalous Dirac string phase in periodically driven systems. Nat. Commun. 15, 1144 (2024).
Guo, Q. et al. Experimental observation of non-Abelian topological charges and edge states. Nature 594, 7862 (2021).
Jiang, T. et al. Four-band non-Abelian topological insulator and its experimental realization. Nat. Commun. 12, 6471 (2021).
Wang, Q.-D., Zhu, Y.-Q., Zhu, S.-L. & Zheng, Z. Synthetic non-Abelian topological charges in ultracold atomic gases. Phys. Rev. A 110, 023321 (2024).
Qiu, H., Tong, S., Zhang, Q., Zhang, K. & Qiu, C. Observation of anomalous Floquet non-Abelian topological insulators. Preprint at https://arxiv.org/abs/2508.06818 (2025).
Li, T. & Hu, H. Floquet non-Abelian topological insulator and multifold bulk-edge correspondence. Nat. Commun. 14, 6418 (2023).
Yang, Y. et al. Non-Abelian physics in light and sound. Science 383, eadf9621 (2024).
Wang, X.-M., Xu, J., Wang, X., Li, Z. & Ma, G. Topological braiding of Bloch eigenmodes protected by non-Abelian quaternion invariants. Preprint at https://arxiv.org/abs/2507.01809 (2025).
Alexander, G. P., Chen, B. G.-G., Matsumoto, E. A. & Kamien, R. D. Colloquium: Disclination loops, point defects, and all that in nematic liquid crystals. Rev. Mod. Phys. 84, 497514 (2012).
Volovik, G. & Mineev, V. Investigation of singularities in superfluid He3 in liquid crystals by the homotopic topology methods. Zh. Eksp. Teor. Fiz. 72, 2256–2274 (1977).
Poenaru, V. & Toulouse, G. The crossing of defects in ordered media and the topology of 3-manifolds. J. Phys. Lett. 38, 887–895 (1977).
Liu, K., Nissinen, J., Slager, R.-J., Wu, K. & Zaanen, J. Generalized liquid crystals: giant fluctuations and the vestigial chiral order of I, O and T matter. Phys. Rev. X 6, 041025 (2016).
Guo, C.-X., Chen, S., Ding, K. & Hu, H. Exceptional non-Abelian topology in multiband non-Hermitian systems. Phys. Rev. Lett. 130, 157201 (2023).
Hu, H. & Zhao, E. Knots and non-Hermitian Bloch bands. Phys. Rev. Lett. 126, 010401 (2021).
König, J. L. K., Yang, K., Budich, J. C. & Bergholtz, E. J. Braid-protected topological band structures with unpaired exceptional points. Phys. Rev. Res. 5, L042010 (2023).
Zhang, Q. et al. Observation of acoustic non-Hermitian Bloch braids and associated topological phase transitions. Phys. Rev. Lett. 130, 017201 (2023).
Rudner, M. S., Lindner, N. H., Berg, E. & Levin, M. Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems. Phys. Rev. X 3, 031005 (2013).
Xiao, L. et al. Non-Hermitian bulk-boundary correspondence in quantum dynamics. Nat. Phys. 16, 761 (2020).
Lin, Q. et al. Topological phase transitions and mobility edges in non-Hermitian quasicrystals. Phys. Rev. Lett. 129, 113601 (2022).
Lin, Q. et al. Observation of non-Hermitian topological Anderson insulator in quantum dynamics. Nat. Commun. 13, 3229 (2022).
Schreiber, A. et al. Decoherence and disorder in quantum walks: from ballistic spread to localization. Phys. Rev. Lett. 106, 180403 (2011).
Maczewsky, L. J., Zeuner, J. M., Nolte, S. & Szameit, A. Observation of photonic anomalous Floquet topological insulators. Nat. Commun. 8, 13756 (2017).
Mukherjee, S. et al. Experimental observation of anomalous topological edge modes in a slowly driven photonic lattice. Nat. Commun. 8, 13918 (2017).
Wintersperger, K. et al. Realization of an anomalous Floquet topological system with ultracold atoms. Nat. Phys. 16, 1058 (2020).
Zhan, X., Qu, D. & Xue, P. Deterministic realization of trace-preserving channels in linear-optical systems. Phys. Rev. A 108, 032611 (2023).
Xu, X.-Y. et al. Measuring the winding number in a large-scale chiral quantum walk. Phys. Rev. Lett. 120, 260501 (2018).
Bong, K.-W. et al. Strong unitary and overlap uncertainty relations: theory and experiment. Phys. Rev. Lett. 120, 230402 (2018).
Li, T. & Lin, Q. Code and dataset for ‘Simulation of a Floquet non-Abelian topological insulator with photonic quantum walk’. Zenodo https://doi.org/10.5281/zenodo.18264637 (2026).