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

Ipsita Mandal

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 $\propto ε^{1/k}$, 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 $\sim ε^{1/2}$ 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 C- and P-symmetric models, establishing the existence of EP$_4$s with $\sim ε^{1/4}$ singularities.

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 order (EPs) 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, we show that the upper- Hessenberg structure of the perturbation directly dictates the leading-order eigenvalue- and eigenvector-splitting to be , 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 EPs in P- and C-symmetric systems are restricted to at most branch points, while PT-symmetric systems generically support EPs with the strongest possible singularities (viz. ). 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 EPs with singularities.

Paper Structure

This paper contains 10 sections, 23 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Perturbation around an exceptional degeneracy
  3. Review of exceptional points in two-band models
  4. Exceptional singularities in three-band models
  5. P-symmetric systems
  6. C-symmetric systems
  7. PT-symmetric systems
  8. Discussions and future outlook
  9. P-symmetric $4 \times 4$ matrix
  10. C-symmetric $4 \times 4$ matrix

Figures (3)

  • Figure 1: Third-order exceptional degeneracies for the lattice models represented by Eq. \ref{['eqhopf-lat']}. (a) 3 pairs of EC$_3$ curves show up on setting $\Delta =-1$ and $a =0.75$ in $\mathcal{H}^{\rm hopf1}_3$. (b) 3 pairs of exceptional surfaces show up on setting $a = 1$ in $\mathcal{H}^{\rm hopf2}_3$.
  • Figure 2: A third-order exceptional line (EC$_3$) for a 3-band model, exemplified by Eq. \ref{['eqtsm']}, showing square-root singularities.
  • Figure 3: Uusing $a=1.25$ in Eq. \ref{['eqPTev']}: (a) The surfaces arising from $p=0$ (red) and $q=0$ (light yellow), which intersect to give a knotted curve (green) representing EC$_3$. (b) The surface of ES$_2$ representing $p^3 + q^2 = 0$ (orange), on which the curve of EC$_3$ (green) is superimposed.