Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

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.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Prices vary by article type



Prices may be subject to local taxes which are calculated during checkout

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


  1. Atiyah, M. The Geometry and Physics of Knots (Cambridge Univ. Press, 1990).

  2. Leach, J., Dennis, M. R., Courtial, J. & Padgett, M. J. Knotted threads of darkness. Nature 432, 165–165 (2004).

    Article  ADS  CAS  PubMed  Google Scholar 

  3. Kedia, H., Bialynicki-Birula, I., Peralta-Salas, D. & Irvine, W. T. M. Tying knots in light fields. Phys. Rev. Lett. 111, 150404 (2013).

    Article  ADS  PubMed  CAS  Google Scholar 

  4. Shimokawa, K., Ishihara, K., Grainge, I., Sherratt, D. J. & Vazquez, M. FtsK-dependent XerCDdif recombination unlinks replication catenanes in a stepwise manner. Proc. Natl Acad. Sci. USA 110, 20906–20911 (2013).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  5. Wojcik, C. C., Sun, X.-Q., Bzdušek, T. & Fan, S. Homotopy characterization of non-Hermitian Hamiltonians. Phys. Rev. B 101, 205417 (2020).

    Article  ADS  CAS  Google Scholar 

  6. Lustig, E. et al. Photonic topological insulator in synthetic dimensions. Nature 567, 356–360 (2019).

    Article  ADS  CAS  PubMed  Google Scholar 

  7. Yuan, L., Lin, Q., Xiao, M. & Fan, S. Synthetic dimension in photonics. Optica 5, 1396–1405 (2018).

    Article  ADS  CAS  Google Scholar 

  8. Adams, C. The Knot Book (American Mathematical Society, 2004).

  9. Thomson, W. II. On vortex atoms. London Edinburgh Phil. Mag. J. Sci. 34, 15–24 (1867).

    Article  Google Scholar 

  10. Pisanty, E. et al. Knotting fractional-order knots with the polarization state of light. Nat. Photon. 13, 569–574 (2019).

    Article  ADS  CAS  Google Scholar 

  11. Pisanty, E. et al. Conservation of torus-knot angular momentum in high-order harmonic generation. Phys. Rev. Lett. 122, 203201 (2019).

    Article  ADS  CAS  PubMed  Google Scholar 

  12. Lian, B., Vafa, C., Vafa, F. & Zhang, S. -C. Chern-Simons theory and Wilson loops in the Brillouin zone. Phys. Rev. B 95, 094512 (2017).

    Article  ADS  Google Scholar 

  13. Sun, X.-Q., Lian, B. & Zhang, S. -C. Double helix nodal line superconductor. Phys. Rev. Lett. 119, 147001 (2017).

    Article  ADS  PubMed  Google Scholar 

  14. Wu, Q., Soluyanov, A. A. & Bzdušek, T. Non-Abelian band topology in noninteracting metals. Science 365, 1273–1277 (2019).

    Article  MathSciNet  CAS  PubMed  MATH  Google Scholar 

  15. Lee, C. H. et al. Imaging nodal knots in momentum space through topolectrical circuits. Nat. Commun. 11, 4385 (2020).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  16. Witten, E. Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351–399 (1989).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Hu, H. & Zhao, E. Knots and non-Hermitian Bloch bands. Phys. Rev. Lett. 126, 010401 (2021).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  18. Weimann, S. et al. Topologically protected bound states in photonic parity–time-symmetric crystals. Nat. Mater. 16, 433–438 (2017).

    Article  ADS  CAS  PubMed  Google Scholar 

  19. Bandres, M. A. et al. Topological insulator laser: experiments. Science 359, eaar4005 (2018).

    Article  PubMed  CAS  Google Scholar 

  20. Zhou, H. et al. Observation of bulk Fermi arc and polarization half charge from paired exceptional points. Science 359, 1009–1012 (2018).

    Article  ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

  21. Zhao, H. et al. Non-Hermitian topological light steering. Science 365, 1163–1166 (2019).

    Article  ADS  CAS  PubMed  Google Scholar 

  22. Weidemann, S. et al. Topological funneling of light. Science 368, 311–314 (2020).

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  23. Wang, K. et al. Generating arbitrary topological windings of a non-Hermitian band. Science 371, 1240–1245 (2021).

    Article  ADS  CAS  PubMed  Google Scholar 

  24. Kozii, V. & Fu, L. Non-Hermitian topological theory of finite-lifetime quasiparticles: prediction of bulk Fermi arc due to exceptional point. Preprint at (2017).

  25. Gao, T. et al. Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard. Nature 526, 554–558 (2015).

    Article  ADS  CAS  PubMed  Google Scholar 

  26. Li, Z. & Mong, R. S. K. Homotopical characterization of non-Hermitian band structures. Phys. Rev. B 103, 155129 (2021).

    Article  ADS  CAS  Google Scholar 

  27. Boada, O., Celi, A., Latorre, J. I. & Lewenstein, M. Quantum simulation of an extra dimension. Phys. Rev. Lett. 108, 133001 (2012).

    Article  ADS  CAS  PubMed  Google Scholar 

  28. Ozawa, T. & Price, H. M. Topological quantum matter in synthetic dimensions. Nat. Rev. Phys. 1, 349–357 (2019).

    Article  Google Scholar 

  29. Yuan, L., Shi, Y. & Fan, S. Photonic gauge potential in a system with a synthetic frequency dimension. Opt. Lett. 41, 741–744 (2016).

    Article  ADS  PubMed  Google Scholar 

  30. Ozawa, T., Price, H. M., Goldman, N., Zilberberg, O. & Carusotto, I. Synthetic dimensions in integrated photonics: from optical isolation to four-dimensional quantum Hall physics. Phys. Rev. A 93, 043827 (2016).

    Article  ADS  Google Scholar 

  31. Bell, B. A. et al. Spectral photonic lattices with complex long-range coupling. Optica 4, 1433–1436 (2017).

    Article  ADS  CAS  Google Scholar 

  32. Dutt, A. et al. Experimental band structure spectroscopy along a synthetic dimension. Nat. Commun. 10, 3122 (2019).

    Article  ADS  PubMed  PubMed Central  CAS  Google Scholar 

  33. Dutt, A. et al. A single photonic cavity with two independent physical synthetic dimensions. Science 367, 59–64 (2020).

    Article  ADS  CAS  PubMed  Google Scholar 

  34. Wang, K. et al. Multidimensional synthetic chiral-tube lattices via nonlinear frequency conversion. Light Sci. Appl. 9, 132 (2020).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  35. Hu, Y., Reimer, C., Shams-Ansari, A., Zhang, M. & Loncar, M. Realization of high-dimensional frequency crystals in electro-optic microcombs. Optica 7, 1189–1194 (2020).

    Article  ADS  CAS  Google Scholar 

  36. Li, G. et al. Dynamic band structure measurement in the synthetic space. Sci. Adv. 7, eabe4335 (2021).

    Article  ADS  PubMed  PubMed Central  CAS  Google Scholar 

  37. Kawabata, K., Shiozaki, K., Ueda, M. & Sato, M. Symmetry and topology in non-Hermitian physics. Phys. Rev. 9, 041015 (2019).

    Article  CAS  Google Scholar 

  38. Gong, Z. et al. Topological phases of non-Hermitian systems. Phys. Rev. 8, 031079 (2018).

    Article  CAS  Google Scholar 

  39. Lee, T. E. Anomalous edge state in a non-Hermitian lattice. Phys. Rev. Lett. 116, 133903 (2016).

    Article  ADS  PubMed  CAS  Google Scholar 

  40. Shen, H., Zhen, B. & Fu, L. Topological band theory for non-Hermitian Hamiltonians. Phys. Rev. Lett. 120, 146402 (2018).

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  41. Okuma, N., Kawabata, K., Shiozaki, K. & Sato, M. Topological origin of non-Hermitian skin effects. Phys. Rev. Lett. 124, 086801 (2020).

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  42. Yuan, L. et al. Photonic gauge potential in one cavity with synthetic frequency and orbital angular momentum dimensions. Phys. Rev. Lett. 122, 083903 (2019).

    Article  ADS  CAS  PubMed  Google Scholar 

  43. Buddhiraju, S., Dutt, A., Minkov, M., Williamson, I. A. D. & Fan, S. Arbitrary linear transformations for photons in the frequency synthetic dimension. Nat. Commun. 12, 2401 (2021).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  44. Baranov, D. G., Krasnok, A. & Alù, A. Coherent virtual absorption based on complex zero excitation for ideal light capturing. Optica 4, 1457–1461 (2017).

    Article  ADS  Google Scholar 

Download references


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

Authors and Affiliations



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.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

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

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing