PT Symmetry in Classical and Quantum Statistical Mechanics

TL;DR

This paper explores PT-symmetric Hamiltonians in classical and quantum statistical mechanics, revealing their real partition functions and complex correlation behaviors, and demonstrating their relevance to models with complex weights and the sign problem.

Contribution

It introduces PT symmetry as a framework for analyzing non-Hermitian models in statistical mechanics, expanding understanding of their properties and solvability.

Findings

Partition functions are real but not necessarily positive.

Correlation functions exhibit sinusoidal modulation with or without decay.

PT-symmetric models include complex magnetic field spin models and QCD at non-zero chemical potential.

Abstract

PT-symmetric Hamiltonians and transfer matrices arise naturally in statistical mechanics. These classical and quantum models often require the use of complex or negative weights and thus fall outside of the conventional equilibrium statistical mechanics of Hermitian systems. PT-symmetric models form a natural class where the partition function is necessarily real, but not necessarily positive. The correlation functions of these models display a much richer set of behaviors than Hermitian systems, displaying sinusoidally-modulated exponential decay, as in a dense fluid, or even sinusoidal modulation without decay. Classical spin models with PT symmetry include Z(N) models with a complex magnetic field, the chiral Potts model and the anisotropic next-nearest-neighbor Ising (ANNNI) model. Quantum many-body problems with a non-zero chemical potential have a natural PT-symmetric representation related to the sign problem. Two-dimensional QCD with heavy quarks at non-zero chemical potential can be solved by diagonalizing an appropriate PT-symmetric Hamiltonian.