Quantum phase transitions mediated by clustered non-Hermitian degeneracies

TL;DR

This paper introduces and analyzes quantum phase transitions mediated by clustered non-Hermitian degeneracies (exceptional points) with higher geometric multiplicity, providing solvable models and a classification scheme for these phenomena.

Contribution

It fills the gap in understanding EP-mediated phase transitions with K>1 by proposing and analyzing a family of exactly solvable models exhibiting clustering.

Findings

Identified and characterized EP clustering with K>1 in quantum systems.

Provided a classification scheme for models based on N partitioning.

Demonstrated solvability of models with multiparametric antisymmetric perturbations.

Abstract

A broad family of phase transitions in the closed as well as open quantum systems is known to be mediated by a non-Hermitian degeneracy (a.k.a. exceptional point, EP) of the Hamiltonian. In the EP limit, in general, the merger of an $N-$plet of the energy eigenvalues is accompanied by a parallel (though not necessarily complete) degeneracy of eigenstates (forming an EP-asociated $K-$plet; in mathematics, $K$ is called the geometric multiplicity of the EP). In the literature, unfortunately, only the benchmark matrix models with $K=1$ can be found. In our paper the gap is filled: the EP-mediated quantum phase transitions with $K>1$ are called "clustered", and a family of benchmark models admitting such a clustering phenomenon is proposed and described. For the sake of maximal simplicity our attention is restricted to the real perturbed-harmonic-oscillator-type N by N matrix Hamiltonians which are exactly solvable and in which the perturbation is multiparametric (i.e., maximally variable) and antisymmetric (i.e., maximally non-Hermitian). A labeling (i.e., an exhaustive classification) of these models is provided by a specific partitioning of N.