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Non-Hermitian swallowtail catastrophe revealing transitions among diverse topological singularities


Exceptional points are a unique feature of non-Hermitian systems at which the eigenvalues and corresponding eigenstates of a Hamiltonian coalesce. Many intriguing physical phenomena arise from the topology of exceptional points, such as bulk Fermi arcs and the braiding of eigenvalues. Here we report that a structurally richer degeneracy morphology, known as the swallowtail catastrophe in singularity theory, can naturally exist in non-Hermitian systems with both parity–time and pseudo-Hermitian symmetries. For the swallowtail, three different types of singularity exist at the same time and interact with each other—an isolated nodal line, a pair of exceptional lines of order three and a non-defective intersection line. Although these singularities seem independent, they are stably connected at a single point—the vertex of the swallowtail—through which transitions can occur. We implement such a system in a non-reciprocal circuit and experimentally observe the degeneracy features of the swallowtail. Based on the frame rotation and deformation of eigenstates, we further demonstrate that the various transitions are topologically protected.

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Fig. 1: Degeneracy features of eigenvalues on various cross-sections in parameter space, showing a swallowtail structure.
Fig. 2: Experimental realization of the swallowtail catastrophe with a non-reciprocal circuit system.
Fig. 3: Experimental observation of the swallowtail catastrophe with the circuit system.
Fig. 4: Understanding the transition of double EL3s to the NIL and NL from eigenframe rotation and deformation.

Data availability

Source data for Fig. 3 are provided with this paper, and the datasets generated and analysed to support this study are available at

Code availability

The code used for calculation and data processing for this paper is available at


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

    Google Scholar 

  2. 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  MATH  Google Scholar 

  3. Kawabata, K., Bessho, T. & Sato, M. Classification of exceptional points and non-Hermitian topological semimetals. Phys. Rev. Lett. 123, 066405 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  4. Miri, M. A. & Alu, A. Exceptional points in optics and photonics. Science 363, eaar7709 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  5. Bergholtz, E. J., Budich, J. C. & Kunst, F. K. Exceptional topology of non-Hermitian systems. Rev. Mod. Phys. 93, 015005 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  6. Tang, W. et al. Exceptional nexus with a hybrid topological invariant. Science 370, 1077–1080 (2020).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  8. Zhang, R. Y., et al. Symmetry-protected topological exceptional chains in non-Hermitian crystals. Preprint at (2022).

  9. Delplace, P., Yoshida, T. & Hatsugai, Y. Symmetry-protected multifold exceptional points and their topological characterization. Phys. Rev. Lett. 127, 186602 (2021).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  11. Xiao, Y. X. et al. Exceptional points make an astroid in non-Hermitian Lieb lattice: evolution and topological protection. Phys. Rev. B 102, 245144 (2020).

    Article  ADS  Google Scholar 

  12. Sayyad, S., Stalhammar, M., Rodland, L. & Kunst, F. K. Symmetry-protected exceptional and nodal points in non-Hermitian systems. Preprint at (2022).

  13. Wan, X., Turner, A. M., Vishwanath, A. & Savrasov, S. Y. Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B 83, 205101 (2011).

    Article  ADS  Google Scholar 

  14. Young, S. M. & Kane, C. L. Dirac semimetals in two dimensions. Phys. Rev. Lett. 115, 126803 (2015).

    Article  ADS  Google Scholar 

  15. Chiu, C. K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016).

    Article  ADS  Google Scholar 

  16. Jia, H. et al. Observation of chiral zero mode in inhomogeneous three-dimensional Weyl metamaterials. Science 363, 148–151 (2019).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  20. Helbig, T. et al. Generalized bulk–boundary correspondence in non-Hermitian topolectrical circuits. Nat. Phys. 16, 747–750 (2020).

    Article  Google Scholar 

  21. Yao, S. & Wang, Z. Edge states and topological invariants of non-Hermitian systems. Phys. Rev. Lett. 121, 086803 (2018).

    Article  ADS  Google Scholar 

  22. Zhong, Q. et al. Sensing with exceptional surfaces in order to combine sensitivity with robustness. Phys. Rev. Lett. 122, 153902 (2019).

    Article  ADS  Google Scholar 

  23. Zhang, X. et al. Experimental observation of an exceptional surface in synthetic dimensions with magnon polaritons. Phys. Rev. Lett. 123, 237202 (2019).

    Article  ADS  Google Scholar 

  24. Okugawa, R. & Yokoyama, T. Topological exceptional surfaces in non-Hermitian systems with parity–time and parity–particle-hole symmetries. Phys. Rev. B 99, 041202 (2019).

    Article  ADS  Google Scholar 

  25. Kim, Y. et al. Dirac line nodes in inversion-symmetric crystals. Phys. Rev. Lett. 115, 036806 (2015).

    Article  ADS  Google Scholar 

  26. Fang, C. et al. Topological nodal line semimetals with and without spin–orbital coupling. Phys. Rev. B 92, 081201 (2015).

    Article  ADS  Google Scholar 

  27. Ahn, J. et al. Band topology and linking structure of nodal line semimetals with Z2 monopole charges. Phys. Rev. Lett. 121, 106403 (2018).

    Article  ADS  Google Scholar 

  28. Arnol’d, V. I. Catastrophe Theory (Springer Science & Business Media, 2003).

  29. Chandrasekaran, A., Shtyk, A., Betouras, J. J. & Chamon, C. Catastrophe theory classification of fermi surface topological transitions in two dimensions. Phys. Rev. Res. 2, 013355 (2020).

    Article  Google Scholar 

  30. Yuan, N. F. Q. & Fu, L. Classification of critical points in energy bands based on topology, scaling, and symmetry. Phys. Rev. B 101, 125120 (2020).

    Article  ADS  Google Scholar 

  31. Kirillov, O. N. & Overton, M. Robust stability at the swallowtail singularity. Front. Phys. 1, 24 (2013).

    Article  Google Scholar 

  32. Raz, O., Pedatzur, O., Bruner, B. D. & Dudovich, N. Spectral caustics in attosecond science. Nat. Photon. 6, 170–173 (2012).

    Article  ADS  Google Scholar 

  33. Gajer, P. The intersection Dold–Thom theorem. Topology 35, 939–967 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  34. Freedman, D. Z. & Van Proeyen, A. Supergravity (Cambridge Univ. Press, 2012).

  35. Soleymani, S. et al. Chiral and degenerate perfect absorption on exceptional surfaces. Nat. Commun. 13, 599(2022).

    Article  ADS  Google Scholar 

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This work was supported by the Research Grants Council of Hong Kong through grants AoE/P-502/20, 16307821, KAUST20SC01 (to C.T.C.), and 16307621 (to H.J.). Y.Z. acknowledges financial support from National Natural Science Foundation of China grant 11701263. We acknowledge Z. Lei for helpful comments in constructing the theoretical framework. Y.Z. thanks P. Feng for assistance with visualizing various geometric configurations.

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Authors and Affiliations



H.J. and C.T.C. planned the project. J.H., Y.W., H.J. and C.T.C. designed the sample. J.H. carried out the measurements. J.H. and H.J. analysed the data. R.-Y.Z., Y.Z. and H.J. constructed the theoretical framework. J.H., R.-Y.Z., Y.Z., H.J. and C.T.C. wrote the paper. J.H., R.-Y.Z., X.O., Y.Z., H.J. and C.T.C. contributed to the discussion.

Corresponding authors

Correspondence to Yifei Zhu, Hongwei Jia or Che Ting Chan.

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Supplementary Information

Supplementary Figs. 1–12, Discussion and Tables 1 and 2.

Supplementary Video 1

Swallowtail under different views.

Supplementary Video 2

Loop deformation process without changing topology.

Source data

Source Data Fig. 3

Source data for Fig. 3.

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Hu, J., Zhang, RY., Wang, Y. et al. Non-Hermitian swallowtail catastrophe revealing transitions among diverse topological singularities. Nat. Phys. (2023).

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