Critical Temperature and Energy Gap for the BCS Equation

TL;DR

This paper derives bounds on the critical temperature and energy gap in the BCS theory, revealing their exponential dependence on interaction strength and a universal ratio between their coefficients.

Contribution

It provides explicit bounds and formulas for $T_c$ and $\\Xi$ in the BCS model, including a universal ratio, applicable across different densities and potentials.

Findings

$T_c$ and $\Xi$ scale as $\exp(-B/\lambda)$ at weak coupling.

The ratio of their coefficients is a universal constant.

Formulas reduce to known low-density results involving scattering length.

Abstract

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