Vortex patterns in Ginzburg-Landau minimizers

Sylvia Serfaty, Etienne Sandier

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

We present a survey of results obtained with Etienne Sandier on vortices in the minimiz- ers of the 2D Ginzburg-Landau energy of superconductivity with an applied magnetic field, in the asymptotic regime of large kappa where vortices become point-like. We de- scribe results which characterize the critical values of the applied field for which vortices appear, their numbers and locations. If the applied field is large enough, it is observed in experiments that vortices are densely packed and form triangular (hexagonal) lattices named Abrikosov lattices. Part of our results is the rigorous derivation of a mean field model describing the optimal density of vortices at leading order in the energy, and then the derivation of a next order limiting energy which governs the positions of the vortices after blow-up at their inter-distance scale. This limiting energy is a logarithmic- type interaction between points in the plane. Among lattice configurations it is uniquely minimized by the hexagonal lattice, thus providing a first justification of the Abrikosov lattice in this regime.

Original languageEnglish (US)
Title of host publicationXVIth International Congress on Mathematical Physics
PublisherWorld Scientific Publishing Co.
Pages246-264
Number of pages19
ISBN (Electronic)9789814304634
ISBN (Print)981430462X, 9789814304627
DOIs
StatePublished - Jan 1 2010

Fingerprint

Ginzburg-Landau
Minimizer
Vortex
vortices
Hexagonal Lattice
Energy
Limiting
derivation
Point Vortex
Mean-field Model
Triangular Lattice
Superconductivity
energy
Justification
Blow-up
Critical value
Logarithmic
Magnetic Field
superconductivity
Configuration

Keywords

  • Abrikosov lattice
  • Gamma-convergence
  • Ginzburg-landau
  • Superconductivity
  • Vortices

ASJC Scopus subject areas

  • Mathematics(all)
  • Physics and Astronomy(all)

Cite this

Serfaty, S., & Sandier, E. (2010). Vortex patterns in Ginzburg-Landau minimizers. In XVIth International Congress on Mathematical Physics (pp. 246-264). World Scientific Publishing Co.. https://doi.org/10.1142/9789814304634_0014

Vortex patterns in Ginzburg-Landau minimizers. / Serfaty, Sylvia; Sandier, Etienne.

XVIth International Congress on Mathematical Physics. World Scientific Publishing Co., 2010. p. 246-264.

Research output: Chapter in Book/Report/Conference proceedingChapter

Serfaty, S & Sandier, E 2010, Vortex patterns in Ginzburg-Landau minimizers. in XVIth International Congress on Mathematical Physics. World Scientific Publishing Co., pp. 246-264. https://doi.org/10.1142/9789814304634_0014
Serfaty S, Sandier E. Vortex patterns in Ginzburg-Landau minimizers. In XVIth International Congress on Mathematical Physics. World Scientific Publishing Co. 2010. p. 246-264 https://doi.org/10.1142/9789814304634_0014
Serfaty, Sylvia ; Sandier, Etienne. / Vortex patterns in Ginzburg-Landau minimizers. XVIth International Congress on Mathematical Physics. World Scientific Publishing Co., 2010. pp. 246-264
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