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

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.

Title changed, to more accurately reflect content. Introduction expanded. References added

References in corpus (13)

• Making Sense of Non-Hermitian Hamiltonians
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Cited by in corpus (16)

• Quantum Classical Correspondence for a non-Hermitian Bose-Hubbard Dimer
• On complexified mechanics and coquaternions
• $\mathcal{P}\mathcal{T}$-symmetric wave guide system with evidence of a third-order exceptional point
• Coupling approach for the realization of a $\mathcal{P}\mathcal{T}$-symmetric potential for a Bose-Einstein condensate in a double well
• Eternal life of entropy in non-Hermitian quantum systems
• Vortices in Bose-Einstein condensates with $\mathcal{P}\mathcal{T}$-symmetric gain and loss
• Numerical calculation of the complex berry phase in non-Hermitian systems
• Realizing PT-symmetric BEC subsystems in closed hermitian systems
• Bifurcations and exceptional points in a PT-symmetric dipolar Bose-Einstein condensate
• Embedding quantum systems with a non-conserved probability in classical environments
• Three perspectives on entropy dynamics in a non-Hermitian two-state system
• Quantum thermodynamics aspects with a thermal reservoir based on $\mathcal{P}\mathcal{T}$-symmetric Hamiltonians
• The Four Dimensional Dirac Equation in Five Dimensions
• Phase transitions and thermodynamic cycles in the broken PT-regime
• Purity oscillations in coupled Bose-Einstein condensates
• Computing eigenfunctions and eigenvalues of boundary value problems with the orthogonal spectral renormalization method