Another operator-theoretical proof for the second-order phase transition in the BCS-Bogoliubov model of superconductivity

TL;DR

This paper presents a simplified operator-theoretical proof that the phase transition in the BCS-Bogoliubov superconductivity model is of second order, by analyzing the differentiability of the gap equation's solutions.

Contribution

It introduces a weaker condition ensuring a unique, twice-differentiable solution and provides an alternative proof of the second-order phase transition in the model.

Findings

Existence of a unique, twice-differentiable solution on a specified interval.

The phase transition in the BCS-Bogoliubov model is confirmed to be of second order.

The proof applies to a generalized potential, not just a constant.

Abstract

In the preceding papers, imposing certain complicated and strong conditions, the present author showed that the solution to the BCS-Bogoliubov gap equation in superconductivity is twice differentiable only on the neighborhoods of absolute zero temperature and the transition temperature so as to show that the phase transition is of the second order from the viewpoint of operator theory. Instead, we impose a certain simple and weak condition in this paper, and show that there is a unique nonnegative solution and that the solution is indeed twice differentiable on a closed interval from a certain positive temperature to the transition temperature as well as pointing out several properties of the solution. We then give another operator-theoretical proof for the second-order phase transition in the BCS-Bogoliubov model. Since the thermodynamic potential has the squared solution in its form, we deal with the squared BCS-Bogoliubov gap equation. Here, the potential in the BCS-Bogoliubov gap equation is a function and need not be a constant.