# Formation of Exceptional Points in pseudo-Hermitian Systems

## TL;DR

This paper investigates how exceptional points emerge in pseudo-Hermitian systems, revealing that their formation depends on a topological index that classifies energy levels, with detailed analysis in a specific $ extit{PT}$-symmetric spin chain model.

## Contribution

It introduces a topological $ ext{Z}_2$ index to characterize energy levels and explains how it governs the formation of second- and third-order exceptional points in pseudo-Hermitian systems.

## Key findings

- Second-order EPs occur only between levels with opposite $ ext{Z}_2$ indices.
- The topological index predicts the emergence of third-order EPs.
- Explicit analysis in a $ extit{PT}$-symmetric spin chain model confirms theoretical predictions.

## Abstract

Motivated by the recent growing interest in the field of $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian systems we theoretically study the emergency of singularities called Exceptional Points ($\textit{EP}$s) in the eigenspectrum of pseudo-Hermitian Hamiltonian as the strength of Hermiticity-breaking terms turns on. Using general symmetry arguments, we characterize the separate energy levels by a topological $\mathbb{Z}_2$ index which corresponds to the signs $\pm 1$ of the eigenvalues of pseudo-metric operator $\hat \zeta$ in the absence of Hermiticity-breaking terms. After that, we show explicitly that the formation of second-order $\textit{EP}$s is governed by this $\mathbb{Z}_2$-index: only the pairs of levels with $\textit{opposite}$ index can provide second-order $\textit{EP}$s. Our general analysis is accompanied by a detailed study of $\textit{EP}$s appearance in an exemplary $\mathcal{P}\mathcal{T}$-symmetric pseudo-Hermitian system with parity operator in the role of $\hat \zeta$: a transverse-field Ising spin chain with a staggered imaginary longitudinal field. Using analytically computed parity indices of all the levels, we analyze the eigenspectrum of the model in general, and the formation of third-order $\textit{EP}$s in particular