Formation of exceptional points in pseudo-Hermitian systems

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