In recent years, exceptional degeneracies have attracted a great deal of interest. The most studied (and simplest) ones are isolated exceptional points. In contrast, exceptional surfaces involve much more complex types of degeneracies. They feature prominently and pose a fundamental theoretical challenge, as physicists’ research scope expands beyond traditional Hermitian systems to non-Hermitian systems. A main goal is to address open quantum systems that carry energy exchanges with the surrounding environment via the imaginary part of their eigen-energies. Remarkably, these exceptional surfaces afford numerous new singularities, such as high-order exceptional points (or lines) appearing as cusps, as well as non-defective degeneracies that are intersections of exceptional surfaces.

Dr. Yifei Zhu, Assistant Professor of the Department of Mathematics at the Southern University of Science and Technology (SUSTech), and his research group in cooperation with physicists from The Hong Kong University of Science and Technology (HKUST) and Xiangtan University, have recently made advances in studying non-Hermitian systems. They discovered that the swallowtail catastrophe, a mathematically celebrated structure from the study of dynamical systems, emerged as a distinguished and generic hypersurface singularity which naturally existed in non-Hermitian systems with parity–time symmetry together with a pseudo-Hermitian symmetry.

As a highlighting point of their work, this novel degeneracy feature of swallowtail catastrophe exhibits the coexistence and intriguing interactions of a diverse range of non-isolated singularity types in energy band structures, including notably cusps and local complete intersections, which had never been observed or analyzed before. Arguably the most challenging part for this research was to analyze such non-isolated singularities using physically meaningful tools from mathematics, which were fundamentally different from those applied to isolated singularities.

Their research work, entitled “Non-Hermitian swallowtail catastrophe revealing transitions among diverse topological singularities”, has been published in Nature Physics, a top journal covering all areas of physics.

Dr. Yifei Zhu’s group and collaborators reported on the swallowtail catastrophe in singularity theory, which could naturally reside in the moduli spaces of non-Hermitian systems with parity–time and pseudo-Hermitian symmetries. The swallowtail contains three distinct types of singular lines which exist and interact simultaneously. Although these singularities appear to be independent, they are stably connected by the swallowtail at a point through which the transitions occur.

The researchers implemented such a system in a nonreciprocal circuit and experimentally observed the degeneracy characteristics of the swallowtail model. By analyzing frame rotation and deformation of the eigenstates, they further demonstrated that the various transitions were topologically protected, using novel mathematical tools from homotopy theory.

According to their research, the topology of these singularities leads to many fascinating physical phenomena, such as bulk Fermi arcs and eigenvalue weaving. Understanding the topology may help further investigate physical phenomena important both theoretically and in applications, such as the bulk–edge correspondence. This manifests the deep interplay between topology and physics.

Jing Hu and Ruo-Yang Zhang from the Department of Physics at HKUST are the first authors of this article. Dr. Yifei Zhu is one of the corresponding authors, and Pingyao Feng, a junior student of Dr. Zhu’s research group, assisted with visualizing various geometric configurations and is among the researchers acknowledged in the article. She provided the image for this news.

This work was partly supported by the National Natural Science Foundation of China (NSFC).

As a commentary and introduction accompanying the publication of this research, Nature Physics has also invited Savannah Garmon of the Osaka Metropolitan University and University of Tokyo to write a New & Views article “Symmetry gives rise to an elegant catastrophe”.

Dr. Yifei Zhu’s current research concentrates on interactions of algebraic topology with algebraic geometry and number theory, especially moduli spaces from spectral algebraic geometry in the context of the Langlands program. He also works on the applications of topology and geometry to interdisciplinary research, including condensed-matter physics and materials science, time series analysis, human–computer interaction, and systems science.

Article link: https://www.nature.com/articles/s41567-023-02048-w

New & Views article link: https://www.nature.com/articles/s41567-023-02055-x