An operator-theoretical proof for the second-order phase transition in the BCS-Bogoliubov model of superconductivity (final version)

TL;DR

This paper proves that the transition to superconductivity in the BCS-Bogoliubov model is a second-order phase transition using operator theory, providing explicit expressions for the specific heat gap at the transition temperature.

Contribution

It offers a rigorous operator-theoretic proof of the second-order phase transition and derives explicit formulas for the specific heat gap in the BCS-Bogoliubov model.

Findings

Confirmed second-order phase transition in the model.

Derived explicit expression for the specific heat gap.

Proved differentiability and uniqueness of the gap solution.

Abstract

We show that the transition from a normal conducting state to a superconducting state is a second-order phase transition in the BCS-Bogoliubov model of superconductivity from the viewpoint of operator theory. Here we have no magnetic field. Moreover we obtain the exact and explicit expression for the gap in the specific heat at constant volume at the transition temperature. To this end, we have to differentiate the thermodynamic potential with respect to the temperature two times. Since there is the solution to the BCS-Bogoliubov gap equation in the form of the thermodynamic potential, we have to differentiate the solution with respect to the temperature two times. Therefore, we need to show that the solution to the BCS-Bogoliubov gap equation is differentiable with respect to the temperature two times as well as its existence and uniqueness. We carry out its proof on the basis of fixed point theorems.