Translation-invariant quasi-free states for fermionic systems and the BCS approximation

TL;DR

This paper rigorously analyzes translation-invariant quasi-free states in fermionic systems with interactions, showing that direct and exchange energies mainly renormalize the chemical potential, thus justifying parts of the BCS approximation and providing bounds on the superfluid transition temperature.

Contribution

It provides a rigorous justification of the BCS approximation for fermionic systems with short-range interactions, including bounds on the critical temperature for superfluidity.

Findings

Direct and exchange energies only renormalize the chemical potential.

The BCS properties are preserved under certain short-range interactions.

Bounds on the critical temperature for superfluidity are established.

Abstract

We study translation-invariant quasi-free states for a system of fermions with two-particle interactions. The associated energy functional is similar to the BCS functional but includes also direct and exchange energies. We show that for suitable short-range interactions, these latter terms only lead to a renormalization of the chemical potential, with the usual properties of the BCS functional left unchanged. Our analysis thus represents a rigorous justification of part of the BCS approximation. We give bounds on the critical temperature below which the system displays superfluidity.