Mean-field theory of a quasi-one-dimensional superconductor in a high   magnetic field
  At high magnetic field, the semiclassical approximation which underlies the Ginzburg-Landau (GL) theory of the mixed state of type II superconductors breaks down. In a quasi-1D superconductor (weakly coupled chains system) with an {\it open Fermi surface}, a high magnetic field stabilizes a cascade of superconducting phases which ends in a strong reentrance of the superconducting phase. The superconducting state evolves from a triangular Abrikosov vortex lattice in the semiclassical regime towards a Josephson vortex lattice in the reentrant phase. We study the properties of these superconducting phases from a microscopic model in the mean-field approximation. The critical temperature is calculated in the quantum limit approximation (QLA) where only Cooper logarithmic singularities are retained while less divergent terms are ignored. The effects of Pauli pair breaking (PPB) and impurity scattering are taken into account. The Gor'kov equations are solved in the same approximation but ignoring the PPB effect. We derive the GL expansion of the free energy. We obtain the specific heat jump at the transition, the sign of the magnetization and the quasi-particle excitation spectrum. The calculation is extended beyond the QLA taking into account all the pairing channels and the validity of the QLA is discussed in detail. 
N. Dupuis