Formation of Exceptional Points in pseudo-Hermitian Systems

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 ($\text{EP}$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 $\text{EP}$s is governed by this ${\mathbb{Z}}_2$-index: only the pairs of levels with $\text{opposite}$ index can provide second-order $\text{EP}$s. Our general analysis is accompanied by a detailed study of $\text{EP}$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 $\text{EP}$s in particular

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