Puiseux series about exceptional singularities dictated by symmetry-allowed Hessenberg forms of perturbation matrices

TL;DR

This paper develops a framework using Puiseux series to analyze exceptional points in non-Hermitian systems, revealing how symmetry constraints influence the order and nature of eigenvalue singularities.

Contribution

It introduces a systematic method linking Hessenberg matrix structures to eigenvalue splittings at exceptional points, with applications to symmetry-invariant models and sensor design implications.

Findings

EP3s in P- and C-symmetric systems are limited to rac{1}{2} branch points.

PT-symmetric systems can support EP3s with rac{1}{3} singularities.

Extension to four-band models shows existence of EP4s with rac{1}{4} singularities.

Abstract

We develop a systematic framework for determining the nature of exceptional points of $n^{\rm th}$ order (EP$_n$s) in non-Hermitian (NH) systems, represented by complex square matrices. By expressing symmetry-preserving perturbations in the Jordan-normal basis of the defective matrix at an EP$_n$, we show that the upper-$k$ Hessenberg structure of the perturbation directly dictates the leading-order eigenvalue- and eigenvector-splitting to be $\propto \epsilon^{1/k}$, when expanded in a Puiseux series. Applying this to three-band NH models invariant under parity (P), charge-conjugation (C), or parity-time-reversal (PT), we find that EP$_3$s in P- and C-symmetric systems are restricted to at most $\sim \epsilon^{1/2}$ branch points, while PT-symmetric systems generically support EP$_3$s with the strongest possible singularities (viz. $\sim \epsilon^{1/3}$). We illustrate these results with concrete three-dimensional models in which exceptional curves and surfaces emerge. We further show that fine-tuned perturbations can suppress the leading-order branch point to a less-singular splitting, which have implications for designing direction-dependent EP-based sensors. The appendix extends the analysis to four-band C- and P-symmetric models, establishing the existence of EP$_4$s with $\sim \epsilon^{1/4}$ singularities.