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Unsupervised learning of topological non-Abelian braiding in non-Hermitian bands

Nature Machine Intelligence volume 6, pages 904–910 (2024)Cite this article

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A preprint version of the article is available at arXiv.

Abstract

The topological classification of energy bands has laid the foundation for the discovery of various topological phases of matter in recent decades. While previous work focused on real-energy bands in Hermitian systems, recent studies have shifted attention to the intriguing topology of complex-energy, or non-Hermitian, bands, freeing them from the constraint of energy conservation. For example, the spectral winding of complex-energy bands can give rise to unique topological structures such as braids, holding substantial promise for advancing quantum computing. However, discussions of complex-energy braids have been predominantly limited to the Abelian braid group \({{\mathbb{B}}}_{2}\) owing to its relative simplicity. Identifying topological non-Abelian braiding remains challenging, as it lacks a universally applicable topological invariant for characterization. Here we present a machine learning algorithm for the unsupervised identification of non-Abelian braiding within multiple complex-energy bands. We demonstrate that the results are consistent with Artin’s well-known topological equivalence conditions in braiding. Inspired by these findings, we introduce a winding matrix as a topological invariant for characterizing braiding topology. The winding matrix also reveals the bulk-edge correspondence of non-Hermitian bands with non-Abelian braiding. Finally, we extend our approach to identify non-Abelian braiding topology in two-dimensional and three-dimensional exceptional semimetals and address the unknotting problem in an unsupervised manner.

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Fig. 1: Non-Abelian braiding topology in non-Hermitian bands.
Fig. 2: Unsupervised identification of topological non-Abelian braids.
Fig. 3: 2D exceptional topological semimetal with non-Abelian braiding topology.
Fig. 4: Solving the unknotted problem unsupervisedly.

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Data availability

All the data necessary for reproducing our results are publicly available via GitHub at https://github.com/longyangking/ml_topology_non_Abelian_braiding. An archived version is deposited in the Zenodo database at https://doi.org/10.5281/zenodo.11077148 (ref. 89).

Code availability

All the code necessary for reproducing our results is publicly available via GitHub at https://github.com/longyangking/ml_topology_non_Abelian_braiding. An archived version is deposited in the Zenodo database at https://doi.org/10.5281/zenodo.11077148 (ref. 89).

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Acknowledgements

This research is supported by Singapore National Research Foundation Competitive Research Program under grant no. NRF-CRP23-2019-0007 and Singapore Ministry of Education Academic Research Fund Tier 2 under grant no. MOE-T2EP50123-0007. H.X. acknowledges the support of the start-up fund and the direct grant (grant no. 4053675) from the Chinese University of Hong Kong. Y.L. gratefully acknowledges the support of the Eric and Wendy Schmidt AI in Science Postdoctoral Fellowship, a Schmidt Futures programme. We thank Y. Chong for helpful discussions.

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Authors and Affiliations

  1. Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore

    Yang Long & Baile Zhang

  2. Department of Physics, The Chinese University of Hong Kong, Hong Kong, China

    Haoran Xue

  3. Centre for Disruptive Photonic Technologies, Nanyang Technological University, Singapore, Singapore

    Baile Zhang

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Y.L. conceived the idea. Y.L. did the theoretical analysis and performed the calculations. Y.L. and B.Z. supervised the project. All authors participated in discussions and wrote the paper.

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Correspondence to Yang Long or Baile Zhang.

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Long, Y., Xue, H. & Zhang, B. Unsupervised learning of topological non-Abelian braiding in non-Hermitian bands. Nat Mach Intell 6, 904–910 (2024). https://doi.org/10.1038/s42256-024-00871-1

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