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Topological complex-energy braiding of non-Hermitian bands


Effects connected with the mathematical theory of knots1 emerge in many areas of science, from physics2,3 to biology4. Recent theoretical work discovered that the braid group characterizes the topology of non-Hermitian periodic systems5, where the complex band energies can braid in momentum space. However, such braids of complex-energy bands have not been realized or controlled experimentally. Here, we introduce a tight-binding lattice model that can achieve arbitrary elements in the braid group of two strands 𝔹2. We experimentally demonstrate such topological complex-energy braiding of non-Hermitian bands in a synthetic dimension6,7. Our experiments utilize frequency modes in two coupled ring resonators, one of which undergoes simultaneous phase and amplitude modulation. We observe a wide variety of two-band braiding structures that constitute representative instances of links and knots, including the unlink, the unknot, the Hopf link and the trefoil. We also show that the handedness of braids can be changed. Our results provide a direct demonstration of the braid-group characterization of non-Hermitian topology and open a pathway for designing and realizing topologically robust phases in open classical and quantum systems.

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Fig. 1: Braiding of two non-Hermitian bands.
Fig. 2: Realization of two-band complex-energy braids in a frequency synthetic dimension.
Fig. 3: Braided non-Hermitian bands in relation to loops on Riemann surfaces.
Fig. 4: Complex-energy braids that form non-trivial links and knots.

Data availability

The data that support the findings of this study are available in Figshare at

Code availability

The code that supports the findings of this study is available in Figshare at


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We thank D. A. B. Miller for providing laboratory space and equipment and X.-Q. Sun for discussions. This work is supported by a MURI project from the US Air Force Office of Scientific Research (grant no. FA9550-18-1-0379), and by a Vannevar Bush Faculty Fellowship from the US Department of Defense (grant no. N00014-17-1-3030).

Author information




K.W., C.C.W. and S.F. conceived the study; K.W. and C.C.W. developed the theory and performed numerical simulations; K.W. and A.D. performed the experiments and processed experimental data. All authors discussed the results and contributed to writing the manuscript. S.F. supervised the work.

Corresponding author

Correspondence to Shanhui Fan.

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The authors declare no competing interests.

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Peer review information Nature thanks Biao Yang and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

This file contains Supplementary Sections 1–4, including notes on the theoretical details, experimental details, extra experimental results, proposals for future experiments and Supplementary Figs. 1–13. See contents page for details.

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Wang, K., Dutt, A., Wojcik, C.C. et al. Topological complex-energy braiding of non-Hermitian bands. Nature 598, 59–64 (2021).

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