The BCS Energy Gap at Low Density

TL;DR

This paper derives the asymptotic behavior of the BCS energy gap at low density, revealing a universal ratio with the critical temperature for certain potentials, advancing understanding of superconductivity in dilute regimes.

Contribution

It provides a precise asymptotic formula for the BCS energy gap at low density and establishes its universal ratio with the critical temperature for a class of potentials.

Findings

Energy gap asymptotically behaves as $ o 0$ with a specific exponential form.

The ratio of energy gap to critical temperature is a universal constant.

Results apply to potentials with negative scattering length and no bound states.

Abstract

We show that the energy gap for the BCS gap equation is $ \Xi = \mu \left( 8 e^{-2} + o(1)\right) \exp\left( \frac{\pi}{2\sqrt{\mu} a}\right) $ in the low density limit $\mu \to 0$. Together with the similar result for the critical temperature [arXiv:0803.3324] this shows that, in the low density limit, the ratio of the energy gap and critical temperature is a universal constant independent of the interaction potential $V$. The results hold for a class of potentials with negative scattering length $a$ and no bound states.