Universal behavior of the BCS energy gap

Abstract

We consider the BCS energy gap $\Xi (T)$ (essentially given by $\Xi (T)\approx \Delta (T,\sqrt{\mu })$, the BCS order parameter) at all temperatures $0\le T\le T_{\text{c}}$ up to the critical one, $T_{\text{c}}$, and show that, in the limit of weak coupling, the ratio $\Xi (T)/T_{\text{c}}$ is given by a universal function of the relative temperature $T/T_{\text{c}}$. On the one hand, this recovers a recent result by Langmann and Triola [Phys. Rev. B 108 (2023), no. 10, article no. 104503] on three-dimensional $s$-wave superconductors for temperatures bounded uniformly away from $T_{\text{c}}$. On the other hand, our result lifts these restrictions, as we consider arbitrary spatial dimensions $d\in \{1,2,3\}$, discuss superconductors with non-zero angular momentum (primarily in two dimensions), and treat the perhaps physically most interesting (due to the occurrence of the superconducting phase transition) regime of temperatures close to $T_{\text{c}}$.