Stability of Abrikosov lattices under gauge-periodic perturbations

TL;DR

This paper proves the asymptotic stability of Abrikosov vortex lattices with arbitrary shapes under gauge-periodic perturbations near the second critical magnetic field, challenging the belief that only triangular lattices are stable.

Contribution

It establishes the stability criteria for Abrikosov lattices of arbitrary shape under gauge-periodic perturbations near H_{c2}.

Findings

Lattices are stable for kappa^2 > (1/2)(1 - 1/beta(tau)).

Lattices are unstable for kappa^2 < (1/2)(1 - 1/beta(tau)).

Results challenge the common belief about stability only for triangular lattices.

Abstract

We consider Abrikosov-type vortex lattice solutions of the Ginzburg-Landau equations of superconductivity, consisting of single vortices, for magnetic fields below but close to the second critical magnetic field H_{c2} = kappa^2 and for superconductors filling the entire R^2. Here kappa is the Ginzburg-Landau parameter. The lattice shape, parameterized by tau, is allowed to be arbitrary (not just triangular or rectangular). Within the context of the time-dependent Ginzburg-Landau equations, called the Gorkov-Eliashberg-Schmidt equations, we prove that such lattices are asymptotically stable under gauge periodic perturbations for kappa^2 > (1/2)(1 - (1/beta(tau)) and unstable for kappa^2 < (1/2)(1 - (1/beta(tau)), where beta(tau) is the Abrikosov constant depending on the lattice shape tau. This result goes against the common belief among physicists and mathematicians that Abrikosov-type vortex lattice solutions are stable only for triangular lattices and kappa^2 > 1/2. (There is no real contradiction though as we consider very special perturbations.)