Critical Temperature and Energy Gap for the BCS Equation

Abstract

We derive upper and lower bounds on the critical temperature $T_c$ and the energy gap $Ξ$ (at zero temperature) for the BCS gap equation, describing spin 1/2 fermions interacting via a local two-body interaction potential $λV(x)$. At weak coupling $λ\ll 1$ and under appropriate assumptions on $V(x)$, our bounds show that $T_c \sim A \exp(-B/λ)$ and $Ξ\sim C \exp(-B/λ)$ for some explicit coefficients $A$, $B$ and $C$ depending on the interaction $V(x)$ and the chemical potential $μ$. The ratio $A/C$ turns out to be a universal constant, independent of both $V(x)$ and $μ$. Our analysis is valid for any $μ$; for small $μ$, or low density, our formulas reduce to well-known expressions involving the scattering length of $V(x)$.