Classification of multiple arbitrary-order non-Hermitian singularities

TL;DR

This paper classifies the topological structures of Riemann surfaces generated by multiple higher-order exceptional points in non-Hermitian systems, revealing permutation classes linked to these singularities with applications in optical microcavities.

Contribution

It introduces a comprehensive classification scheme for the topology of Riemann surfaces arising from multiple arbitrary-order exceptional points, including permutation classes and their building blocks.

Findings

All permutation classes of holonomy matrices are categorized.

Permutation classes are derived from cyclic building blocks.

Results are verified using non-Hermitian Hamiltonians and optical microcavities.

Abstract

We demonstrate general classifications of Riemann surface topology generated by multiple arbitrary-order exceptional points of quasi-stationary states. Our studies reveal all possible product permutations of holonomy matrices that describe a stroboscopic encircling of 2nd order exceptional points. The permutations turn out to be categorized into a finite number of classes according to the topological structures of the Riemann surfaces. We further show that the permutation classes can be derived from combinations of cyclic building blocks associated with higher-order exceptional points. Our results are verified by an effective non-Hermitian Hamiltonian founded on generic Jordan forms and then examined in physical systems of desymmetrized optical microcavities.