Gap generation in the BCS model with finite range temporal interaction

TL;DR

This paper rigorously proves that the BCS model with finite range temporal interaction becomes equivalent to the mean field BCS model in the thermodynamic limit when the interaction range is sufficiently long, using a perturbation expansion.

Contribution

It provides a rigorous proof of the equivalence between the finite range temporal interaction BCS model and the mean field BCS model in the thermodynamic limit.

Findings

Equivalence holds when the interaction range is sufficiently long.

A uniformly convergent perturbation expansion was used.

The result confirms the mean field approximation validity in this regime.

Abstract

In the [BCS] paper the theory of superconductivity was developed for the BCS model, in which the (instantaneous) interaction is only between fermions of opposite momentum and spin. Such model was analyzed by variational methods, finding that a superconducting behavior is energetically favorable. Subsequently it was claimed that in the thermodynamic limit the BCS model is equivalent to the (exactly solvable) quadratic mean field BCS model; a rigorous proof of this claim is however still lacking. In this paper we consider the BCS model with a finite range temporal interaction, and we prove rigorously its equivalence with the mean field BCS model in the thermodinamic limit if the range is long enough, by a (uniformly convergent) perturbation expansion about mean field theory.