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 Hc2 = κ 2 and for superconductors filling the entire R 2 . Here κ is the Ginzburg-Landau parameter. The lattice shape, parameterized 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-Schmidt equations, we prove that such lattices are asymptotically stable under gauge periodic perturbations for κ 2 > 1 2 (1 − 1 β(τ ) ) and unstable for κ 2 < 1 2 (1 − 1 β(τ ) ), 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 κ 2 > 1 2 . (There is no real contradiction though as we consider very special perturbations.) *

Magnetic Vortices, Abrikosov Lattices, and Automorphic Functions

“…The stability of Abrikosov lattices was shown in [77] for gauge periodic perturbations, i.e. perturbations having the same translational lattice symmetry as the solutions themselves, and in [78] for local, more precisely, H 1 , perturbations.…”

On stability of Abrikosov vortex lattices

“…. 3843 vq =0 0.25 1.000 26 1.002 26 1.008 01 1.003 30 1.024 28 1.073 64 1.012 19 1.072 60 1.189 62 1.026 59 1.13093 1.305 52 1.061 25 1.229 95 1.47005 1.180 34 1.424 80 1.732 61 1.000 22 1. 002 20 1.…”

“…Remark 1.9. Relation (1.49) for Abrikosov lattices was used in [52], where it played an important role. This condition is well known in algebraic geometry and number theory (see e.g.…”

“…Abrikosov lattices have been extensively studied in the physics literature. Among many rigorous results, we mention that the existence of these solutions was proven rigorously in [41,10,17,7,60,52]. Moreover, important and fairly detailed results on asymptotic behaviour of solutions, for κ → ∞ and applied magnetic fields, h, satisfying h ≤ 1 2 ln κ+const (the London limit), were obtained in [8] (see this paper and the book [48] for references to earlier work).…”

Multi-Vortex Non-radial Solutions to the Magnetic Ginzburg-Landau Equations

“…Recently, Sigal and Tzanateas have found Abrikosov type lattice solutions with infinite energy on the whole plane [25] and have proven these lattice solutions are stable under gauge periodic perturbations [26].…”