Characterizing all non-Hermitian degeneracies using algebraic approaches: Defectiveness and asymptotic behavior

TL;DR

This paper provides a comprehensive algebraic framework to characterize all types of non-Hermitian degeneracies, including their coalescence and response to perturbations, with applications demonstrated through various examples.

Contribution

It introduces a systematic algebraic approach to analyze multi-block degeneracies and their asymptotic behavior in non-Hermitian systems, expanding understanding beyond traditional degeneracies.

Findings

Unified algebraic characterization of all non-Hermitian degeneracies

Demonstrated analysis of degeneracy coalescence and perturbation response

Validated approach through diverse experimental examples

Abstract

The presence of degeneracies plays a crucial role in describing the behavior of non-Hermitian (NH) systems. In these systems, there are two key types of degeneracies: $n$-bolical degeneracies, which are analogous to Hermitian degeneracies, and various forms of exceptional points, each associated with different orders that correspond to sizes of the Jordan blocks. These types of degeneracies may coalesce at the same energy level, forming multi-block degeneracies. To understand how a multi-block degenerate NH system responds to perturbations, one should address how each types of involved degeneracies disperse. In this work, we systematically characterize the asymptotic behavior of all types of multi-block degeneracies in NH systems using a rigorous mathematical formulation. Through a range of examples, we demonstrate that our algebraic approach can facilitate the analysis of NH degeneracies in various settings relevant to experiments.