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 close to the second critical magnetic field Hc2 = κ2 and for superconductors filling the entire . Here κ is the Ginzburg–Landau parameter. The lattice shape, parametrized by τ, is allowed to be arbitrary (not just triangular or rectangular). Within the context of the time-dependent Ginzburg–Landau equations, called the Gorkov–Eliashberg–Schmid equations, we prove that such lattices are asymptotically stable under gauge-periodic perturbations for and unstable for , where β(τ) is the Abrikosov constant depending on the lattice shape τ. This result goes against the common belief among physicists and mathematicians that Abrikosov-type vortex lattice solutions are stable only for triangular lattices and . (There is no real contradiction though as we consider very special perturbations.)