Non-Hermitian N-state degeneracies: unitary realizations via antisymmetric anharmonicities

TL;DR

This paper explores the degeneracy phenomena in quantum systems with non-Hermitian Hamiltonians, providing new models for N-state degeneracies at exceptional points and classifying these processes based on spectral partitioning.

Contribution

It introduces non-numerical models for N>2 and K>1 degeneracies in non-Hermitian quantum systems, expanding beyond the well-studied two-state case.

Findings

Constructed benchmark models for N>2 degeneracies

Classified degeneracy processes via spectral partitioning

Demonstrated feasibility of non-numerical modeling for complex degeneracies

Abstract

The phenomenon of degeneracy of an $N-$plet of bound states is studied in the framework of quantum theory of closed (i.e., unitary) systems. For an underlying Hamiltonian $H=H(\lambda)$ the degeneracy occurs at a Kato's exceptional point $\lambda^{(EPN)}$ of order $N$ and of the spectral geometric multiplicity $K<N$. In spite of the phenomenological appeal of the concept (tractable as a quantum phase transition, or as a unitary processes of the loss of the observability of the system), the dedicated literature deals, predominantly, just with the models where $N=2$ and $K=1$. In our paper it is shown that the construction of the $N>2$ and $K>1$ benchmark models of the process of degeneracy becomes feasible and non-numerical for a broad class of specific, maximally non-Hermitian anharmonic-oscillator toy-model Hamiltonians. An exhaustive classification of non-equivalent processes is given by a partitioning of the unperturbed spectrum into equidistant and centered unperturbed subspectra.