On Abrikosov Lattice Solutions of the Ginzburg-Landau Equation

Abstract

Building on earlier work, we have given in [29] a proof of existence of Abrikosov vortex lattices in the Ginzburg-Landau model of superconductivity and shown that the triangular lattice gives the lowest energy per lattice cell. After [29] was published, we realized that it proves a stronger result than was stated there. This result is recorded in the present paper. The proofs remain the same as in [29], apart from some streamlining.

On Abrikosov Lattice Solutions of the Ginzburg-Landau Equations

“…For a lattice L ⊂ R 2 , we denote by Ω L and |Ω L | the basic lattice cell and its area, respectively. The following results were proven in [31,32]:…”

“…As was mentioned above, we revisit the existence proof of [31,32] streamlining some arguments and providing some essential details either missing or only briefly mentioned ( [31,32]) in earlier proofs of the existence of Abrikosov vortex lattices.…”

“…We will use the following preposition, first used by [25] and proved in [30] (an alternate proof is given in in Appendix A of [32]). Here c s satisfies the condition c s+t − c s − c t − 1 2 bs ∧ t ∈ 2πZ.…”

“…The above investigation was completed and extended in [31,32]. To formulate the results of these papers, we introduce some notation and definitions.…”

“…where β(L) is the Abrikosov parameter, see e.g. [31,32]. For a lattice L ⊂ R 2 , we denote by Ω L and |Ω L | the basic lattice cell and its area, respectively.…”