Linear Criterion for an Upper Bound on the Bardeen-Cooper-Schrieffer Critical Temperature

TL;DR

This paper introduces a linear criterion to estimate an upper bound on the critical temperature for superconductivity, providing insights into the conditions where superconductivity can or cannot occur.

Contribution

It establishes a new linear criterion for an upper temperature bound in superconductivity, extending the understanding beyond the existing lower-bound criterion.

Findings

Identifies a temperature $T_u$ above which superconductivity does not occur.

Shows that $T_u$ can be greater than, equal to, or less than the lower bound $T_l$ depending on the system.

Estimates $T_u$ for half-spaces, demonstrating exponential smallness in the weak coupling limit.

Abstract

Since Bardeen-Cooper-Schrieffer theory of superconductivity is non-linear, it is difficult to study superconducting properties analytically. There is a more tractable linear criterion which determines a temperature $T_l$ below which the system is superconducting. Here, we observe that there is a similar linear criterion which gives a temperature $T_u$ above which no superconductivity occurs. We provide examples of translation invariant systems where $T_u>T_l$ as well as systems where $T_u=T_l$. Furthermore, we estimate $T_u$ for half-spaces and show that it is exponentially small in the weak coupling limit, exhibiting the same asymptotics as the critical temperature for full space.