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

TL;DR

The paper develops and analyzes two linear criteria for the BCS critical temperature, $T_l$ and $T_u$, defined via spectral thresholds of the operators $L_T-V$ and $D_T-V$, and connects them to the standard $K_T-V$ criterion in the zero-total-momentum sector. It proves that in translation-invariant, SU(2)-invariant settings these criteria can coincide or differ (with explicit examples where $T_u>T_l$), and it provides the first rigorous upper bounds for Tc in systems with a boundary (half-spaces), showing that boundary effects on Tc vanish in the weak-coupling limit and that superconductivity on half-spaces persists only at temperatures exponentially small in the coupling, identical in leading order to the full-space result. The analysis hinges on a careful Birman-Schwinger reduction, relating $L_T$, $D_T$ to $K_T$, and exploiting spectral monotonicity and asymptotics in one, two, and three dimensions. In particular, the work delivers weak- and strong-coupling asymptotics for $T_l$ and $T_u$ in 1D, establishes conditions for the uniqueness of Tc in translation-invariant cases, and extends boundary-related Tc comparisons to the upper bound, thereby clarifying the role of geometry in superconductivity within the BCS framework.

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.

Figures (1)

• Figure 1: Two circles of radius $\sqrt{\mu}$, centered at $(-\vert q \vert,0)$ and $(\vert q \vert,0)$. Assume that the coordinate system is chosen such that $q=(|q|,0,0)$ and write the vector $p=(p_1, \tilde{p})$. The points in $A_2$ lie in one of the circles, but not in the other. Thus $A_2$ is the shaded area. The points in $A_3$ either lie in both circles or outside both of them, i.e. the white area. The sketch is adapted from roos_bcs_2023.

Theorems & Definitions (23)

• Lemma 1.1
• Lemma 1.2
• Lemma 1.4
• Remark 1.5
• Remark 1.6
• Theorem 1.7
• Remark 1.8
• Remark 1.9
• Theorem 1.10
• Remark 1.11