Low-temperature breakdown of many-body perturbation theory for thermodynamics

TL;DR

This paper demonstrates that finite-temperature many-body perturbation theory fails to converge at zero temperature when ground state degeneracies change, due to the nonanalytic nature of the Boltzmann factor, revealing a fundamental flaw.

Contribution

It analytically and numerically shows the nonconvergence of perturbation theory at zero temperature caused by degeneracy lifting and nonanalytic Boltzmann factors, challenging previous assumptions.

Findings

Perturbation theory diverges at zero temperature when degeneracies change.

Nonanalytic Boltzmann factor causes nonconvergence at T=0.

The flaw is fundamental and unlikely to be fixed within the perturbation framework.

Abstract

It is shown analytically and numerically that the finite-temperature many-body perturbation theory in the grand canonical ensemble has zero radius of convergence at zero temperature when the energy ordering or degree of degeneracy for the ground state changes with the perturbation strength. When the degeneracy of the reference state is partially or fully lifted at the first-order Hirschfelder-Certain degenerate perturbation theory, the grand potential and internal energy diverge as $T \to 0$. Contrary to earlier suggestions of renormalizability by the chemical potential $\mu$, this nonconvergence, first suspected by W. Kohn and J. M. Luttinger, is caused by the nonanalytic nature of the Boltzmann factor $e^{-E/k_\text{B}T}$ at $T=0$, also plaguing the canonical ensemble, which does not involve $\mu$. The finding reveals a fundamental flaw in perturbation theory, which is deeply rooted in the mathematical limitation of power-series expansions and is unlikely to be removed within its framework.