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

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.