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

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.

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Cited by in corpus (7)

• Finite-temperature many-body perturbation theory for electrons: Algebraic recursive definitions, second-quantized derivation, linked-diagram theorem, general-order algorithms, grand canonical and canonical ensembles
• Nonconvergence of the Feynman-Dyson diagrammatic perturbation expansion of propagators
• General solution to the Kohn-Luttinger nonconvergence problem
• Finite-temperature many-body perturbation theory for vibrations: Recursions, algebraic reduction, second-quantized reduction, diagrammatic rules, linked-diagram theorem, finite-temperature self-consistent field, and general-order algorithm
• Thermal mean-field theories
• Thermal quasi-particle theory
• Complex analysis of divergent perturbation theory at finite temperature