Phase space representation of non-Hermitian system with $\mathcal{PT}$-symmetry

TL;DR

This paper explores the phase space behavior of non-Hermitian, $ ext{PT}$-symmetric systems using the Wigner distribution, revealing a second-order phase transition at the exceptional point.

Contribution

It introduces a phase space framework for $ ext{PT}$-symmetric non-Hermitian systems and derives a generalized continuity equation for the Wigner function flow.

Findings

Identifies a second-order phase transition at the exceptional point.

Derives a generalized continuity equation for Wigner function flow.

Analyzes circulation values in phase space.

Abstract

We present a phase space study of non-Hermitian Hamiltonian with $\mathcal{PT}$-symmetry based on the Wigner distribution function. For an arbitrary complex potential, we derive a generalized continuity equation for the Wigner function flow and calculate the related circulation values. Studying vicinity of an exceptional point, we show that a $\mathcal{PT}$-symmetric phase transition from an unbroken $\mathcal{PT}$-symmetry phase to a broken one is a second-order phase transition.