On Abrikosov Lattice Solutions of the Ginzburg-Landau Equations

Abstract

We prove existence of Abrikosov vortex lattice solutions of the Ginzburg-Landau equations of superconductivity, with multiple magnetic flux quanta per a fundamental cell. We also revisit the existence proof for the Abrikosov vortex lattices, streamlining some arguments and providing some essential details missing in earlier proofs for a single magnetic flux quantum per a fundamental cell. 1 The Ginzburg-Landau theory is reviewed in every book on superconductivity and most of the books on solid state or condensed matter physics. For reviews of rigorous results see the papers [11,12,20,29] and the books [28,17,21,27] 2 In the problem we consider here it is appropriate to deal with Helmholtz free energy at a fixed average magnetic field b := 1|Ω| Ω curl A, where |Ω| is the area or volume of Ω.

Vortex lattice solutions of the ZHK Chern–Simons equations

“…The proof of the following statement follows the argument in section 7 of [31], see also the proof of proposition 6.2 of [10].…”

“…In this section, adopting the strategy from [9] (see also [10,31]), we replace the original system of equations (2.12), by one giving the same solutions and which is easier to handle. We begin with de ning appropriate spaces we work in.…”

“…The ZHK Chern-Simons model explains the FQHE and related phenomena at the main lling fractions ν = 1/2k + 1 and allows a reconstruction, through 'deformation quantization' (as de ned in [17]), of the Laughlin's many-particle variational state and its re nements due to Halperin explaining lling fractions other than ν = 1/2k + 1 (see [30,35] for reviews) 10 .…”

Vortex lattice solutions of the ZHK Chern-Simons equations

“…In this section, adopting the strategy from [9] (see also [30,10]), we replace the original system of equations, (2.12), by one giving the same solutions and which is easier to handle. We begin with defining appropriate spaces we work in.…”

Microscopic Derivation of Ginzburg-Landau Theory and the BCS Critical Temperature Shift in a Weak Homogeneous Magnetic Field

“…Their theory is macroscopic in nature and contains no reference to a microscopic mechanism behind the phenomenon of superconductivity. The GL equations show a rich mathematical structure, which has been investigated in great detail, see, e.g., [7,8,45,44,10,11] and references therein, and they also inspired interesting new concepts beyond the realm of their original application.…”