Ginzburg-Landau model with small pinning domains

TL;DR

This paper analyzes how small impurity regions in a superconductor influence vortex positions in the Ginzburg-Landau model, showing vortices tend to localize within pinning domains and are governed by a local energy minimization.

Contribution

It introduces a detailed analysis of vortex pinning in the Ginzburg-Landau model with small impurity domains, linking vortex locations to a finite-dimensional energy minimization problem.

Findings

Vortices are confined within pinning domains for small v.

Vortex degrees are equal to 1 inside pinning domains.

Vortex positions are determined by a local renormalized energy.

Abstract

We consider a Ginzburg-Landau type energy with a piecewise constant pinning term $a$ in the potential $(a^2 - |u|^2)^2$. The function $a$ is different from 1 only on finitely many disjoint domains, called the {\it pinning domains}. These pinning domains model small impurities in a homogeneous superconductor and shrink to single points in the limit $\v\to0$; here, $\v$ is the inverse of the Ginzburg-Landau parameter. We study the energy minimization in a smooth simply connected domain $\Omega \subset \mathbb{C}$ with Dirichlet boundary condition $g$ on $\d \O$, with topological degree ${\rm deg}_{\d \O} (g) = d >0$. Our main result is that, for small $\v$, minimizers have $d$ distinct zeros (vortices) which are inside the pinning domains and they have a degree equal to 1. The question of finding the locations of the pinning domains with vortices is reduced to a discrete minimization problem for a finite-dimensional functional of renormalized energy. We also find the position of the vortices inside the pinning domains and show that, asymptotically, this position is determined by {\it local renormalized energy} which does not depend on the external boundary conditions.