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

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 ($\propto ε$) in the Jordan-normal basis of the defective matrix at an EP$_n$, we show that the upper-$\ell$ Hessenberg structure of the perturbation matrix directly dictates the leading-order eigenvalue- and eigenvector-splitting to be $\propto ε^{1/\ell}$, 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 $\sim ε^{1/2}$ branch points, while PT-symmetric systems generically support EP$_3$s with the strongest possible singularities (viz. $\sim ε^{1/3}$). 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 P- and C-symmetric models, establishing the existence of EP$_4$s with $\sim ε^{1/4}$ and $\sim ε^{1/2}$ singularities, respectively.