Critical Temperature and Energy Gap for the BCS Equation

Details Critical Temperature and Energy Gap for the BCS Equation Critical Temperature and Energy Gap for the BCS Equation

2008-01-27

arxiv.org/abs/0801.4159v2

Phys. Rev. B 77, 184517 (2008)

Physics

Condensed Matter

Superconductivity

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

Scientific paper

10.1103/PhysRevB.77.184517

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 $V(x)$.

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