An operator-theoretical study of the specific heat and the critical magnetic field in the BCS-Bogoliubov model of superconductivity

TL;DR

This paper uses operator theory to analyze the temperature dependence of specific heat and critical magnetic field in the BCS-Bogoliubov superconductivity model, revealing universal constants and smooth behavior.

Contribution

It provides a rigorous operator-theoretic analysis of the temperature dependence of key superconducting properties, including explicit formulas and universality results.

Findings

The solution to the gap equation is independent of specific superconductors.

The critical magnetic field varies smoothly with temperature.

The ratio of specific heats is a universal constant.

Abstract

In the preceding paper, introducing a cutoff, the present author gave a proof of the statement that the transition to a superconducting state is a second-order phase transition in the BCS-Bogoliubov model of superconductivity on the basis of fixed-point theorems, and solved the long-standing problem of the second-order phase transition from the viewpoint of operator theory. In this paper we study the temperature dependence of the specific heat and the critical magnetic field in the model from the viewpoint of operator theory. We first show some properties of the solution to the BCS-Bogoliubov gap equation with respect to the temperature, and give the exact and explicit expression for the gap in the specific heat divided by the specific heat. We then show that it does not depend on superconductors and is a universal constant. Moreover, we show that the critical magnetic field is smooth with respect to the temperature, and point out the behavior of both the critical magnetic field and its derivative.