Complex analysis of divergent perturbation theory at finite temperature

TL;DR

This paper uses complex analysis to study the convergence of finite-temperature perturbation theory, revealing how zeros of the partition function cause divergences and how these relate to quantum phase transitions.

Contribution

It provides a detailed mathematical analysis of the divergence mechanisms in finite-temperature perturbation theory and links these to quantum phase transition representations.

Findings

Zeros of the partition function lead to poles in internal energy.

Higher temperatures increase the radius of convergence.

Degeneracy causes zero-temperature divergence.

Abstract

We investigate the convergence properties of finite-temperature perturbation theory by considering the mathematical structure of thermodynamic potentials using complex analysis. We discover that zeros of the partition function lead to poles in the internal energy and logarithmic singularities in the Helmholtz free energy which create divergent expansions in the canonical ensemble. Analysing these zeros reveals that the radius of convergence increases for higher temperatures. In contrast, when the reference state is degenerate, these poles in the internal energy create a zero radius of convergence in the zero-temperature limit. Finally, by showing that the poles in the internal energy reduce to exceptional points in the zero-temperature limit, we unify the two main mathematical representations of quantum phase transitions.