Quantum and Classical Statistical Mechanics of a Class of non-Hermitian Hamiltonians

TL;DR

This paper explores the thermodynamics of non-Hermitian PT-symmetric oscillators with real spectra, using WKB approximation for quantum partition functions and complex phase space integration for classical cases.

Contribution

It introduces a method to analyze thermodynamics of non-Hermitian Hamiltonians with real spectra, combining quantum and classical approaches.

Findings

WKB approximation accurately estimates energy spectra for small quantum numbers.

Quantum partition functions are computed and analyzed across temperature regimes.

Classical partition functions require integration over complex phase space.

Abstract

This paper investigates the thermodynamics of a large class of non-Hermitian, $PT$-symmetric oscillators, whose energy spectrum is entirely real. The spectrum is estimated by second-order WKB approximation, which turns out to be very accurate even for small quantum numbers, and used to generate the quantum partition function. Graphs showing the thermal behavior of the entropy and the specific heat, at all regimes of temperature, are given. To obtain the corresponding classical partition function it turns out to be necessary in general to integrate over a complex "phase space". For the wrong-sign quartic, whose equivalent Hermitian Hamiltonian is known exactly, it is demonstrated explicitly how this formulation arises, starting from the Hermitian case.