# The BCS critical temperature in a weak homogeneous magnetic field

## TL;DR

This paper rigorously demonstrates that a weak homogeneous magnetic field decreases the BCS critical temperature proportionally to the field strength, extending previous physics results with a new phase approximation technique.

## Contribution

It provides a rigorous mathematical derivation of the magnetic field's effect on the BCS critical temperature, generalizing prior physics-based results.

## Key findings

- Critical temperature decreases linearly with magnetic field strength.
- Introduces a rigorous phase approximation method.
- Extends WHH theory results to a mathematical framework.

## Abstract

We show that, within a linear approximation of BCS theory, a weak homogeneous magnetic field lowers the critical temperature by an explicit constant times the field strength, up to higher order terms. This provides a rigorous derivation and generalization of results obtained in the physics literature from WHH theory of the upper critical magnetic field. A new ingredient in our proof is a rigorous phase approximation to control the effects of the magnetic field.