Formation of exceptional points in pseudo-Hermitian systems

Motivated by the recent growing interest in the field of PT-symmetric Hamiltonian systems we theoretically study the emergence of singularities called exceptional points (EPs) in the eigenspectrum of a pseudo-Hermitian Hamiltonian as the strength of Hermiticity-breaking terms increases. Using general symmetry arguments, we characterize the separate energy levels by a topological index which corresponds to the signs +/- 1 of the eigenvalues of pseudometric operator (zeta) over cap. in the absence of Hermiticity-breaking terms. After that, we show explicitly that the formation of second-order EPs is governed by this index: only the pairs of levels with opposite index can provide second-order EPs. Our general analysis is accompanied by a detailed study of EPs' appearance in an exemplary PT -symmetric pseudo-Hermitian system with the parity operator in the role of (zeta) over cap: 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 EPs in particular.