Breakdown of superconductivity in a magnetic field with self-intersecting zero set

TL;DR

This paper proves the monotonicity of the lowest eigenvalue of a magnetic Laplacian with a self-intersecting zero set and applies it to determine the unique transition point from superconducting to normal states in the Ginzburg-Landau model.

Contribution

It establishes the monotonicity of the eigenvalue in magnetic fields with complex zero sets and identifies a unique threshold for superconducting transition.

Findings

Eigenvalue is monotone near infinity of magnetic field strength.

Transition from superconducting to normal occurs at a unique magnetic field threshold.

Application to Ginzburg-Landau model confirms the transition point.

Abstract

We prove that the lowest eigenvalue of the Laplace operator with a magnetic field having a self-intersecting zero set is a monotone function of the parameter defining the strength of the magnetic field, in a neighborhood of infinity. We apply this monotonicity result on the study of the transition from superconducting to normal states for the Ginzburg-Landau model, and prove that the transition occurs at a unique threshold value of the applied magnetic field.