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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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.
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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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.
The authors declare no competing interests.
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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.
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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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