Abrikosov lattice solutions of the Ginzburg-Landau equations

Tzaneteas, Tim; Sigal, Israel Michael

doi:10.1090/conm/535/10542

Cited by 14 publications

(55 citation statements)

References 25 publications

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“…. 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.…”

Section: Problem and Resultsmentioning

confidence: 99%

“…023 70 1.073 02 1.010 89 1.071 05 1.188 27 1. 02401 1.128 43 1.303 64 1.056 28 1.226 21 1.467 61 1.171 95 1.419 71 1.729 61 1.000 18 1.002 02 1.007 69 1.002 34 1.021 96 1.071 17 1.008 94 1.066 42 1.184 23 1.020 15 1.120 89 1.297 94 1.048 90 1.214 89 1.460 24 1.159 60 1.404 24 1.720 53 . Sample [7].…”

Section: Problem and Resultsmentioning

confidence: 99%

“…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).…”

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confidence: 92%

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On stability of Abrikosov vortex lattices

Sigal¹,

Tzaneteas²

2018

Advances in Mathematics

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The Ginzburg-Landau equations play a key role in superconductivity and particle physics. They inspired many imitations in other areas of physics. These equations have two remarkable classes of solutions -vortices and (Abrikosov) vortex lattices. For the standard cylindrical geometry, the existence theory for these solutions, as well as the stability theory of vortices are well developed. The latter is done within the context of the time-dependent Ginzburg-Landau equations -the Gorkov-Eliashberg-Schmid equations of superconductivity -and the abelian Higgs model of particle physics.We study stability of Abrikosov vortex lattices under finite energy perturbations satisfying a natural parity condition (both defined precisely in the text) for the dynamics given by the Gorkov-Eliashberg-Schmid equations. For magnetic fields close to the second critical magnetic field and for arbitrary lattice shapes, we prove that there exist two functions on the space of lattices, such that Abrikosov vortex lattice solutions are asymptotically stable, provided the superconductor is of Type II and these functions are positive, and unstable, for superconductors of Type I, or if one of these functions is negative. 2 κ is a positive constant, called the Ginzburg-Landau parameter, ∇ A = ∇ − iA and ∆ A = ∇ A ·∇ A are the covariant gradient and Laplacian. Physically, |Ψ| 2 gives the (local) density of superconducting electrons (Cooper pairs), B = curl A is the magnetic field. The second equation is Ampère's law with J S = Im(Ψ∇ A Ψ) being the supercurrent associated to the electrons having formed Cooper pairs. We assume, as common, that superconductors fill in all of R 2 (the cylindrical geometry in R 3 ). In this case, curl A :By far, the most important and celebrated solutions of the Ginzburg-Landau equations are magnetic vortex lattice solutions, discovered by Abrikosov ([1]), and known as (Abrikosov) vortex lattice solutions or simply Abrikosov or vortex lattices. Among other things, understanding these solutions is important for maintaining the superconducting current in Type II superconductors, i.e., for κ > 1 √ 2 . 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). Further extensions to the Ginzburg-Landau equations for anisotropic and high temperature superconductors in the κ → ∞ regime can be found in [4,5]. (See [30, 50] for reviews.)In this paper we are interested in dynamics of of the Abrikosov lattices, as described by the time-dependent generalization of the Ginzburg-Landau equations proposed by Schmid ([49]) and Gorkov and Eliashberg ([23]) (earlier versions are due to Bardeen and Stephen and Anderson, Luttinger and Wert...

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Section: Problem and Resultsmentioning

confidence: 99%

Section: Problem and Resultsmentioning

confidence: 99%

mentioning

confidence: 92%

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On stability of Abrikosov vortex lattices

Sigal¹,

Tzaneteas²

2018

Advances in Mathematics

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“…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]:…”

Section: Introductionmentioning

confidence: 91%

“…Assume χ ∈ H 1 loc and is L−periodic (we say, χ ∈ H 1 per ). Following [31], we differentiate the equation E λ (e isχ ψ, α + s∇χ) = E λ (ψ, α), w.r.to s at s = 0, use that curl ∇χ = 0 and integrate by parts, to obtain Re −∆ a n +α ψ + κ 2 (|ψ| 2 − 1)ψ,iχψ + J(ψ, α), ∇χ = 0. (6.10) (Due to conditions (6.3) -(6.4) and the L−periodicity of χ, there are no boundary terms.)…”

Section: Setup Of the Bifurcation Problemmentioning

confidence: 99%

On Abrikosov Lattice Solutions of the Ginzburg-Landau Equations

Chenn

Smyrnelis

Sigal

2018

Math Phys Anal Geom

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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.

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Magnetic Vortices, Abrikosov Lattices, and Automorphic Functions

Sigal

2015

Mathematical and Computational Modeling

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We address the macroscopic theory of superconductivity -the Ginzburg-Landau theory. This theory is based on the celebrated Ginzburg -Landau equations. First developed to explain and predict properties of superconductors, these equations form an integral part -Abelean-Higgs component -of the standard model of particle physics and, in general, have a profound influence on physics well beyond their original designation area.We present recent results and review earlier works involving key solutions of these equations -the magnetic vortices (of Nielsen-Olesen ( Nambu) strings in particle physics) and vortex lattices, their existence, stability and dynamics, and how they relate to the modified theta functions appearing in number theory. Some automorphic functions appear naturally and play a key role in this theory.

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Abrikosov lattice solutions of the Ginzburg-Landau equations

Abstract: Building on the earlier work of Odeh, Barany, Golubitsky, Turski and Lasher we give a proof of existence of Abrikosov vortex lattices in the Ginzburg-Landau model of superconductivity.

Cited by 14 publications

References 25 publications

On stability of Abrikosov vortex lattices

On stability of Abrikosov vortex lattices

On Abrikosov Lattice Solutions of the Ginzburg-Landau Equations

Magnetic Vortices, Abrikosov Lattices, and Automorphic Functions

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