Energy and Vorticity of the Ginzburg-Landau Model with Variable Magnetic Field

TL;DR

This paper analyzes the Ginzburg-Landau model with a smoothly varying magnetic field, deriving asymptotic energy formulas and revealing non-uniform vortex densities in the superconductor.

Contribution

It provides the first detailed asymptotic analysis of the Ginzburg-Landau energy with a variable magnetic field, including vortex distribution insights.

Findings

Derived an asymptotic formula for the energy

Showed energy minimizers contain vortices

Vortex density varies non-uniformly across the sample

Abstract

We consider the Ginzburg-Landau functional with a variable applied magnetic field in a bounded and smooth two dimensional domain. The applied magnetic field varies smoothly and is allowed to vanish non-degenerately along a curve. Assuming that the strength of the applied magnetic field varies between two characteristic scales, and the Ginzburg-Landau parameter tends to $+\infty$, we determine an accurate asymptotic formula for the minimizing energy and show that the energy minimizers have vortices. The new aspect in the presence of a variable magnetic field is that the density of vortices in the sample is not uniform.