### Abstract

We study the Ginzburg-Landau energy for a superconductor submitted to a magnetic field h_{ex} just below the "second critical field" H_{c2}. When the Ginzburg-Landau parameter ε is small, we show that the mean energy per unit volume can be approximated by a reduced energy on a torus. Moreover, we expand this reduced energy in terms of H_{c2} - h_{ex} : when this quantity gets small, the problem amounts to a minimization problem on a finite-dimensional space, equivalent to the "lowest Landau level" in other approaches. The functions in this finite-dimensional space can themselves be expressed via the Jacobi Theta function of a lattice. This connects the Ginzburg-Landau energy to the "Abrikosov problem" of locating vortices optimally on a lattice.

Original language | English (US) |
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Pages (from-to) | 183-202 |

Number of pages | 20 |

Journal | Selecta Mathematica, New Series |

Volume | 13 |

Issue number | 2 |

DOIs | |

State | Published - Sep 1 2007 |

### Keywords

- Abrikosov lattices
- Lowest Landau level
- Second critical field
- Superconductivity
- Theta functions
- Vortices

### ASJC Scopus subject areas

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

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## Cite this

_{c2}.

*Selecta Mathematica, New Series*,

*13*(2), 183-202. https://doi.org/10.1007/s00029-007-0043-7