Clustering of exceptional points and dynamical phase transitions

TL;DR

This paper investigates how exceptional points in non-Hermitian quantum systems lead to phase transitions and eigenfunction behavior changes, revealing a stabilization mechanism and clustering effects causing dynamical phase transitions.

Contribution

It introduces the concept of eigenfunction clustering at exceptional points and their role in dynamical phase transitions in open quantum systems.

Findings

Eigenfunctions become nearly orthogonal at maximum width bifurcation.

Clustering of exceptional points causes a dynamical phase transition.

System stabilization occurs through local structures described by Hermitian operators.

Abstract

The eigenvalues of a non-Hermitian Hamilton operator are complex and provide not only the energies but also the lifetimes of the states of the system. They show a non-analytical behavior at singular (exceptional) points (EPs). The eigenfunctions are biorthogonal, in contrast to the orthogonal eigenfunctions of a Hermitian operator. A quantitative measure for the ratio between biorthogonality and orthogonality is the phase rigidity of the wavefunctions. At and near an EP, the phase rigidity takes its minimum value. The lifetimes of two nearby eigenstates of a quantum system bifurcate under the influence of an EP. When the parameters are tuned to the point of maximum width bifurcation, the phase rigidity suddenly increases up to its maximum value. This means that the eigenfunctions become almost orthogonal at this point. This unexpected result is very robust as shown by numerical results for different classes of systems. Physically, it causes an irreversible stabilization of the system by creating local structures that can be described well by a Hermitian Hamilton operator. Interesting non-trivial features of open quantum systems appear in the parameter range in which a clustering of EPs causes a dynamical phase transition.