Gamma-convergence of 2D Ginzburg-Landau functionals with vortex concentration along curves

TL;DR

This paper analyzes the asymptotic behavior of 2D Ginzburg-Landau functionals with vortex concentration along curves, showing Gamma-convergence to a classical energy and describing vortex distribution via potential theory.

Contribution

It establishes the Gamma-convergence of Ginzburg-Landau functionals with vortex concentration along curves, linking vortex distribution to Green equilibrium measures.

Findings

Vortices concentrate along prescribed curves in the limit.

Energy functionals Gamma-converge to a classical potential theory energy.

Vortex distribution is characterized by Green equilibrium measures.

Abstract

We study the variational convergence of a family of two-dimensional Ginzburg-Landau functionals arising in the study of superfluidity or thin-film superconductivity, as the Ginzburg-Landau parameter epsilon tends to 0. In this regime and for large enough applied rotations (for superfluids) or magnetic fields (for superconductors), the minimizers acquire quantized point singularities (vortices). We focus on situations in which an unbounded number of vortices accumulate along a prescribed Jordan curve or a simple arc in the domain. This is known to occur in a circular annulus under uniform rotation, or in a simply connected domain with an appropriately chosen rotational vector field. We prove that, suitably normalized, the energy functionals Gamma-converge to a classical energy from potential theory. Applied to global minimizers, our results describe the limiting distribution of vortices along the curve in terms of Green equilibrium measures.