Phase Transitions in Quasi-Hermitian Quantum Models at Exceptional Points of Order Four

Phase transition in quantum mechanics is interpreted as an evolution, at the end of which, typically, a parameter-dependent and Hermitizable Hamiltonian H(g) loses its observability. In the language of mathematics, such a “quantum catastrophe” occurs at an exceptional point of order N (EPN). Although the Hamiltonian H(g) itself becomes unphysical in the limit of g→gEPN, it is shown that it can play the role of an unperturbed operator in an innovative perturbation-approximation analysis of the vicinity of the EPN singularity. As long as such an analysis is elementary at N≤3 and purely numerical at N≥5, we pick up N=4 and demonstrate that for an arbitrary quantum system, the specific (i.e., already sufficiently phenomenologically rich) EP4 degeneracy becomes accessible via a unitary evolution process. This process is shown realizable inside a parametric domain Dphysical, the boundaries of which are determined, near gEP4, non-numerically. Possible relevance of such a mathematical result in the context of non-Hermitian photonics is emphasized.