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Experimental observation of non-Abelian topological charges and edge states


In the last few decades, topological phase1,2,3,4,5,6,7,8,9,10,11 has emerged as a new classification of matter states beyond the Ginzburg–Landau symmetry-breaking paradigm. The underlying global invariant is usually well characterized by integers, such as Chern numbers or winding numbers—the Abelian charges12,13,14,15. Very recently, researchers proposed the notion of non-Abelian topological charges16,17,18,19, which possess non-commutative and fruitful braiding structures with multiple (more than one) bandgaps tangled together. Here we experimentally observe the non-Abelian topological charges in a time-reversal and inversion-symmetric transmission line network. The quaternion-valued non-Abelian topological charges are clearly mapped onto an eigenstate-frame sphere. Moreover, we find a non-Abelian quotient relation that provides a global perspective on the distribution of edge/domain-wall states. Our work opens the door towards characterization and manipulation of non-Abelian topological charges, which may lead to interesting observables such as trajectory-dependent Dirac/Weyl node collisions in two-dimensional systems16,17,20, admissible nodal line configurations in three dimensions16,19,20, and may provide insight into certain strongly correlated phases of twisted bilayer graphene21.

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Fig. 1: Non-Abelian topological charges.
Fig. 2: Realizing non-Abelian topological charges in a tight-binding model.
Fig. 3: Experimental characterization of non-Abelian topological charges and hard boundary edge states.
Fig. 4: Non-Abelian quotient relation.

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Data availability

The data and code that support the findings of this study are available in DataSpace@HKUST at


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This work is supported by the Hong Kong RGC (AoE/P-02/12, 16304717, 16310420) and the Hong Kong Scholars Program (XJ2019007). L.Z. acknowledges the National Natural Science Foundation of China (grant no. 12074230) and Shanxi Province 100-Plan Talent Program. S.Z. acknowledges support from the ERC Consolidator Grant (TOPOLOGICAL), the Royal Society and the Wolfson Foundation. C.T.C. thanks R. Cheng for making space available to construct the network.

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Authors and Affiliations



Q.G., B.Y., S.Z. and C.T.C. conceived the idea; T.J. designed the transmission line network with input from Z.-Q.Z. and C.T.C.; Q.G. and T.J. carried out all measurements; Q.G., T.J., R.-Y.Z., Z.-Q.Z., B.Y., S.Z. and C.T.C. developed and carried out the theoretical analysis; L.Z. participated in the analysis and discussion of the results. B.Y., S.Z. and C.T.C. supervised the whole project. Q.G., T.J., R.-Y.Z. and B.Y. wrote the manuscript and the Supplementary Information with input from all other authors.

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Correspondence to Biao Yang, Shuang Zhang or C. T. Chan.

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Peer review information Nature thanks Xiang Ni, Baile Zhang and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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This file contains Supplementary Text and Data, Supplementary Tables 1-3, Supplementary References and Supplementary Figures 1-25.

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Guo, Q., Jiang, T., Zhang, RY. et al. Experimental observation of non-Abelian topological charges and edge states. Nature 594, 195–200 (2021).

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