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 $\Xi$ (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 $V(x)$.

RevTeX4, 23 pages. Revised version, to appear in Phys. Rev. B

References in corpus (1)

• Many-Body Physics with Ultracold Gases

Cited by in corpus (2)

• The BCS Critical Temperature for Potentials with Negative Scattering Length
• The solution to the BCS gap equation and the second-order phase transition in superconductivity