### 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 language | English (US) |
---|---|

Title of host publication | XVIth International Congress on Mathematical Physics |

Publisher | World Scientific Publishing Co. |

Pages | 246-264 |

Number of pages | 19 |

ISBN (Electronic) | 9789814304634 |

ISBN (Print) | 981430462X, 9789814304627 |

DOIs | |

State | Published - Jan 1 2010 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Physics and Astronomy(all)

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*XVIth International Congress on Mathematical Physics.*World Scientific Publishing Co., pp. 246-264. https://doi.org/10.1142/9789814304634_0014

}

TY - CHAP

T1 - Vortex patterns in Ginzburg-Landau minimizers

AU - Serfaty, Sylvia

AU - Sandier, Etienne

PY - 2010/1/1

Y1 - 2010/1/1

N2 - 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.

AB - 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.

KW - Abrikosov lattice

KW - Gamma-convergence

KW - Ginzburg-landau

KW - Superconductivity

KW - Vortices

UR - http://www.scopus.com/inward/record.url?scp=84910037903&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84910037903&partnerID=8YFLogxK

U2 - 10.1142/9789814304634_0014

DO - 10.1142/9789814304634_0014

M3 - Chapter

AN - SCOPUS:84910037903

SN - 981430462X

SN - 9789814304627

SP - 246

EP - 264

BT - XVIth International Congress on Mathematical Physics

PB - World Scientific Publishing Co.

ER -