The order-by-order renormalization of the self-consistent mean-field potential in many-body perturbation theory for normal Fermi systems is investigated in detail. Building on previous work mainly by Balian and de Dominicis, as a key result we derive a thermodynamic perturbation series that manifests the consistency of the adiabatic zero-temperature formalism with perturbative statistical mechanics---for both isotropic and anisotropic systems---and satisfies at each order and for all temperatures the thermodynamic relations associated with Fermi-liquid theory. These properties are proved to all orders.The order-by-order renormalization of the self-consistent mean-field potential in many-body perturbation theory for normal Fermi systems is investigated in detail. Building on previous work mainly by Balian and de Dominicis, as a key result we derive a thermodynamic perturbation series that manifests the consistency of the adiabatic zero-temperature formalism with perturbative statistical mechanics---for both isotropic and anisotropic systems---and satisfies at each order and for all temperatures the thermodynamic relations associated with Fermi-liquid theory. These properties are proved to all orders.

arXiv:1804.03040Physical Review CJun 28, 201910 citations

Perturbation Theory at Finite Temperatures

Wolfgang Nolting

Springer eBooksJan 1, 20180 citations

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

So Hirata

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 second-order grand potential and internal energy diverge as $T\ensuremath{\rightarrow}0$. Contrary to earlier suggestions of renormalizability by the chemical potential $\ensuremath{\mu}$, this nonconvergence, first suspected by Kohn and Luttinger, is caused by the nonanalytic nature of the Boltzmann factor ${e}^{\ensuremath{-}E/{k}_{\text{B}}T}$ at $T=0$ , also plaguing the canonical ensemble, which does not involve $\ensuremath{\mu}$. 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.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 second-order grand potential and internal energy diverge as $T\ensuremath{\rightarrow}0$. Contrary to earlier suggestions of renormalizability by the chemical potential $\ensuremath{\mu}$, this nonconvergence, first suspected by Kohn and Luttinger, is caused by the nonanalytic nature of the Boltzmann factor ${e}^{\ensuremath{-}E/{k}_{\text{B}}T}$ at $T=0$, also plaguing the canonical ensemble, which does not involve $\ensuremath{\mu}$. 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.

arXiv:2006.00078Physical Review AJan 26, 202110 citations

Perturbation Theory at Finite Temperatures

Pure and applied physicsJan 1, 19610 citations

Complex analysis of divergent perturbation theory at finite temperature

Yi Sun, Hugh G. A. Burton

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 that create divergent expansions in the canonical ensemble. Analyzing these zeros reveals that the radius of convergence increases at 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.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 that create divergent expansions in the canonical ensemble. Analyzing these zeros reveals that the radius of convergence increases at 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.

arXiv:2203.02377The Journal of Chemical PhysicsMay 2, 20221 citation

Perturbation Theory at Finite Temperatures

Wolfgang Nolting, William Dean Brewer

Springer eBooksJan 1, 20090 citations

T=0

, also plaguing the canonical ensemble, which does not involve $\ensuremath{\mu}$. 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.