Quasi-one-dimensional superconductors: from weak to strong magnetic field

We discuss the possible existence of a superconducting phase at high magnetic field in organic quasi-one-dimensional conductors. We consider in particular (i) the formation of a Larkin-Ovchinnikov-Fulde-Ferrell state, (ii) the role of a temperature-induced dimensional crossover occuring when the transverse coherence length $\xi_z(T)$ becomes of the order of the lattice spacing, and (iii) the effect of a magnetic-field-induced dimensional crossover resulting from the localization of the wave functions at high magnetic field. In the case of singlet spin pairing, only the combination of (i) and (iii) yields a picture consistent with recent experiments in the Bechgaard salts showing the existence of a high-field superconducting phase. We point out that the vortex lattice is expected to exhibit unusual characteristics at high magnetic field.