The solution to the BCS gap equation and the second-order phase transition in superconductivity

TL;DR

This paper investigates how the solution to the BCS gap equation varies with temperature and demonstrates that the transition to superconductivity is a second-order phase transition, providing a new proof of existence and uniqueness.

Contribution

It offers a new proof of the existence and uniqueness of the BCS gap equation solution and analyzes the temperature dependence, showing the phase transition is second-order.

Findings

Solution varies continuously with temperature

Transition to superconductivity is second-order

Provides a new proof of solution's uniqueness

Abstract

The existence and the uniqueness of the solution to the BCS gap equation of superconductivity is established in previous papers, but the temperature dependence of the solution is not discussed. In this paper, in order to show how the solution varies with the temperature, we first give another proof of the existence and the uniqueness of the solution and point out that the unique solution belongs to a certain set. Here this set depends on the temperature $T$. We define another certain subset of a Banach space consisting of continuous functions of both $T$ and $x$. Here, $x$ stands for the kinetic energy of an electron minus the chemical potential. Let the solution be approximated by an element of the subset of the Banach space above. We second show, under this approximation, that the transition to a superconducting state is a second-order phase transition.