Universality in low-dimensional BCS theory

TL;DR

This paper explores the universal ratio of energy gap to critical temperature in BCS theory, extending the known universality from three dimensions to one and two dimensions, with a focus on the weak coupling limit.

Contribution

It extends the proof of universality in BCS theory to lower dimensions, specifically d=1,2, in the weak coupling regime.

Findings

Universality of the energy gap to critical temperature ratio in 1D and 2D BCS models.

Rigorous proof of this universality in the weak coupling limit.

Extension of previous 3D results to lower dimensions.

Abstract

It is a remarkable property of BCS theory that the ratio of the energy gap at zero temperature $\Xi$ and the critical temperature $T_c$ is (approximately) given by a universal constant, independent of the microscopic details of the fermionic interaction. This universality has rigorously been proven quite recently in three spatial dimensions and three different limiting regimes: weak coupling, low density, and high density. The goal of this short note is to extend the universal behavior to lower dimensions $d=1,2$ and give an exemplary proof in the weak coupling limit.