Ginzburg-Landau minimizers with high topological degrees in an annulus

TL;DR

This paper investigates the asymptotic behavior of Ginzburg-Landau energy minimizers in an annulus, revealing a critical degree for vortex configurations influenced by boundary conditions and the GL parameter.

Contribution

It provides a detailed analysis of vortex structures in annular domains, identifying a critical degree for the emergence of giant vortices and their interaction with boundary vortices.

Findings

Existence of a critical degree |ln ε| for vortex type transition.

Giant vortices form below the critical degree, while mixed vortex configurations occur above.

As the GL parameter tends to zero, vortices tend to the outer boundary.

Abstract

Motivated by recent experiments on fermionic rings, we study the asymptotic behaviour of minimizers of the Ginzburg-Landau (GL) energy in an annulus with a Dirichlet data which depends on the GL parameter on the outer boundary. We show that there is a critical degree of order $|\ln \varepsilon|$ under which the ground state displays a giant vortex and above which minimizers exhibit a combination of a giant vortex and vortices which tend to the outer boundary as the GL parameter tends to zero. Our analysis relies on the construction of suitable upper and lower bounds, on the extension to a slightly bigger annulus and on the minimization of the mean-field energy appearing in the lower bound. In order to be able to derive the minimum of this energy we use the symmetry of the domain and criticality with respect to inner variations.