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Non-Hermitian swallowtail catastrophe revealing transitions among diverse topological singularities

Nature Physics (2023)Cite this article

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Abstract

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 https://drive.google.com/file/d/11nFGtefO8XpxqJ_hm0Ew2hRa8tVdvkl6/view?usp=share_link.

Code availability

The code used for calculation and data processing for this paper is available at https://drive.google.com/file/d/11nFGtefO8XpxqJ_hm0Ew2hRa8tVdvkl6/view?usp=share_link.

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Acknowledgements

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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Author notes

  1. These authors contributed equally: Jing Hu, Ruo-Yang Zhang.

Authors and Affiliations

  1. Department of Physics, Hong Kong University of Science and Technology, Hong Kong, China

    Jing Hu, Ruo-Yang Zhang, Yixiao Wang, Hongwei Jia & Che Ting Chan

  2. School of Materials Science and Engineering, Xiangtan University, Xiangtan, China

    Xiaoping Ouyang

  3. Department of Mathematics, Southern University of Science and Technology, Shenzhen, China

    Yifei Zhu

  4. Institute for Advanced Study, Hong Kong University of Science and Technology, Hong Kong, China

    Hongwei Jia

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Contributions

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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The authors declare no competing interests.

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Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.

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Supplementary information

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). https://doi.org/10.1038/s41567-023-02048-w

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