New research conducted by a team of scientists from the Hong Kong University of Science and Technology, Xiangtan University, and Southern University of Science and Technology has revealed a possible connection between catastrophe theory and non-Hermitian physics. The researchers discovered that a structurally rich degeneracy, known as the swallowtail catastrophe, can naturally occur in non-Hermitian systems.
The study was inspired by previous research that used homotopy theory to classify topological singularities in liquid crystals. The team sought to extend this approach to understand the singularities in non-Hermitian systems. They applied the concept of an eigenvector frame rotation along a loop encircling a singularity to these systems, which posed significant challenges due to the non-orthogonality of eigenvectors.
To tackle these challenges, the researchers collaborated with mathematician Yifei Zhu and explored the occurrence of degeneracies known as exceptional surfaces in non-Hermitian systems. They tracked the zeros in the discriminant of the characteristic polynomial of Hamiltonian matrices under specific symmetries in the parameter space.
The team’s experiments and calculations led to several interesting findings. They observed that distinct degeneracy lines can be stably connected at a single meeting point, forming a unique structure that is symmetry-protected. This structure resembles four swallowtails combined. However, further mathematical tools are needed to demonstrate the topological equivalence of a loop encircling two nodal lines and a loop encircling four exceptional lines of order three.
The researchers established a connection between catastrophe theory and non-Hermitian physics, two previously unrelated areas of study. By using homotopical methods, they gained a topological understanding of non-isolated singularities in non-Hermitian systems.
The swallowtail catastrophe observed in non-Hermitian bands represents a new type of topological gapless phase. The team is currently conducting studies on the bulk-edge correspondence and the unconventional bulk-Fermi arc in this phase.
The findings from this research could have implications for future physics studies and could also pave the way for new research in mathematics. The team plans to further develop the mathematical component of their work in future studies, aiming to uncover mathematically systematic and physically meaningful structures underlying the swallowtail catastrophe. They believe that the algebraic method of intersection homotopy/homology should be further developed as a powerful tool for understanding non-isolated singularities in both physics and mathematics.
Overall, this research has shed light on the existence of the swallowtail catastrophe in non-Hermitian systems, revealing new transitions among diverse topological singularities.
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