### 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) |
---|---|

Pages (from-to) | 183-202 |

Number of pages | 20 |

Journal | Selecta Mathematica, New Series |

Volume | 13 |

Issue number | 2 |

DOIs | |

State | Published - Sep 2007 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Selecta Mathematica, New Series*,

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

**Lowest Landau level approach in superconductivity for the Abrikosov lattice close to Hc2
.** / Aftalion, Amandine; Serfaty, Sylvia.

Research output: Contribution to journal › Article

*Selecta Mathematica, New Series*, vol. 13, no. 2, pp. 183-202. https://doi.org/10.1007/s00029-007-0043-7

}

TY - JOUR

T1 - Lowest Landau level approach in superconductivity for the Abrikosov lattice close to Hc2

AU - Aftalion, Amandine

AU - Serfaty, Sylvia

PY - 2007/9

Y1 - 2007/9

N2 - We study the Ginzburg-Landau energy for a superconductor submitted to a magnetic field hex just below the "second critical field" Hc2. 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 Hc2 - hex : 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.

AB - We study the Ginzburg-Landau energy for a superconductor submitted to a magnetic field hex just below the "second critical field" Hc2. 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 Hc2 - hex : 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.

KW - Abrikosov lattices

KW - Lowest Landau level

KW - Second critical field

KW - Superconductivity

KW - Theta functions

KW - Vortices

UR - http://www.scopus.com/inward/record.url?scp=38749138184&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38749138184&partnerID=8YFLogxK

U2 - 10.1007/s00029-007-0043-7

DO - 10.1007/s00029-007-0043-7

M3 - Article

VL - 13

SP - 183

EP - 202

JO - Selecta Mathematica, New Series

JF - Selecta Mathematica, New Series

SN - 1022-1824

IS - 2

ER -