Stability of Abrikosov lattices under gauge-periodic perturbations

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.)

18 pages

References in corpus (1)

• On Abrikosov Lattice Solutions of the Ginzburg-Landau Equations

Cited by in corpus (6)

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• Magnetic Vortices, Abrikosov Lattices and Automorphic Functions
• Irreducible Ginzburg-Landau fields in dimension 2
• Ginzburg-Landau Equations on Non-compact Riemann Surfaces
• Differential Equations of Quantum Mechanics