Phase transitions in quasi-Hermitian quantum models at exceptional points of order four

Quantum phase transition is interpreted as an evolution, at the end of which a parameter-dependent 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 → g^EPN, it is shown that it can play the role of an unperturbed operator in a perturbation-approximation analysis of the vicinity of the EPN singularity. Such an analysis is elementary at Nleq 3 and numerical at Ngeq 5, so we pick up N=4. We demonstrate that the specific EP4 degeneracy becomes accessible via a unitary evolution process realizable inside a parametric domain {cal D}_{rm physical}, the boundaries of which are determined non-numerically. Possible relevance of such a mathematical result in the context of non-Hermitian photonics is emphasized.