Local minimizers with unbounded vorticity for the $2$d Ginzburg-Landau functional

TL;DR

This paper proves the existence of local minimizers with unbounded vorticity in the 2D Ginzburg-Landau model under high magnetic fields, providing detailed vortex configurations and asymptotics.

Contribution

It introduces a new method to construct local minimizers with many vortices for high magnetic fields, extending previous results to larger vortex numbers.

Findings

Existence of local minimizers with vortex count up to nearly the magnetic field strength.

Precise vortex localization and separation results.

Refined asymptotic descriptions of vortex configurations.

Abstract

A central focus of Ginzburg-Landau theory is the understanding and characterization of vortex configurations. On a bounded domain $\Omega\subseteq \mathbb{R}^2,$ global minimizers, and critical states in general, of the corresponding energy functional have been studied thoroughly in the limit $\epsilon\to 0,$ where $\epsilon>0$ is the inverse of the Ginzburg-Landau parameter. The presence of an applied magnetic field of strength $h_{ex}\gg 1$ makes possible the existence of stable vortex states. A notable open problem is whether there are solutions of the Ginzburg-Landau equation for any number of vortices below $ h_{ex} |\Omega| /2 \pi,$ for external fields of up to super-heating field strength. The best earlier partial results give, for every $0<c<1,$ and $K>0,$ the existence of local minimizers of the Ginzburg-Landau functional with a prescribed number of vortices in the range $1 \leq N \leq \min \{ K | \log \epsilon |, c ( h_{ex} |\Omega| /2 \pi ) \}$ and for values of $1\ll_\epsilon h_{ex}$ smaller than a power of the Ginzburg-Landau parameter. In this paper, we prove that there are constants $K_1, \alpha>0$ such that given natural numbers satisfying \[1\leq N \leq \frac{h_{ex}}{2\pi}(|\Omega|-h_{ex}^{-1/4}),\] local minimizers of the Ginzburg-Landau functional with this many vortices exist, for fields such that $K_1\leq h_{ex} \leq 1/\epsilon^{\alpha}.$ Our strategy consists in combining: the minimization over a subset of configurations for which we can obtain a very precise localization of vortices; expansion of the energy in terms of a modified vortex interaction energy that allows for a reduction to a potential theory problem; and a quantitative vortex separation result for admissible configurations. Our results provide detailed information about the vorticity and refined asymptotics of the local minimizers that we construct.