Coherent-state path integrals in quantum thermodynamics

TL;DR

This work presents a coherent-state path-integral framework for quantum thermodynamics and emphasizes subtle continuum-limit issues that can affect equilibrium results. By carefully handling variable transformations, functional determinants, and Matsubara-summation regularization, the authors demonstrate exact agreement with canonical Hamiltonian results across bosonic/fermionic oscillators, single-site Bose-Hubbard/Hubbard models, weakly interacting Bose gases, and BCS superconductivity, including finite-range interactions. The notes provide a unified, pedagogical treatment of imaginary-time and frequency-space formalisms, illustrating the crucial role of convergence factors and Hubbard-Stratonovich decoupling in obtaining correct thermodynamic quantities. Overall, the work serves as a rigorous reference for applying coherent-state path integrals to quantum many-body thermodynamics, highlighting common pitfalls and practical procedures for consistent results.

Abstract

In these notes, we elucidate some subtle aspects of coherent-state path integrals, focusing on their application to the equilibrium thermodynamics of quantum many-particle systems. These subtleties emerge when evaluating path integrals in the continuum, either in imaginary time or in Matsubara-frequency space. Our central message is that, when handled with due care, the path integral yields results identical to those obtained from the canonical Hamiltonian approach. We illustrate this through a pedagogical treatment of several paradigmatic systems: the bosonic and fermionic harmonic oscillators, the single-site Bose-Hubbard and Hubbard models, the weakly-interacting Bose gas with contact and finite-range interactions, and the BCS superconductor with contact and finite-range interactions.