Inhomogeneous superconductivity in condensed matter and QCD

Inhomogeneous superconductivity arises when the species participating in thepairing phenomenon have different Fermi surfaces with a large enoughseparation. In these conditions it could be more favorable for each of thepairing fermions to stay close to its Fermi surface and, differently from theusual BCS state, for the Cooper pair to have a non zero total momentum. Forthis reason in this state the gap varies in space, the ground state isinhomogeneous and a crystalline structure might be formed. This situation wasconsidered for the first time by Fulde, Ferrell, Larkin and Ovchinnikov, andthe corresponding state is called LOFF. The spontaneous breaking of the spacesymmetries in the vacuum state is a characteristic feature of this phase and isassociated to the presence of long wave-length excitations of zero mass. Thesituation described here is of interest both in solid state and in elementaryparticle physics, in particular in Quantum Chromo-Dynamics at high density andsmall temperature. In this review we present the theoretical approach to theLOFF state and its phenomenological applications using the language of theeffective field theories.Comment: RevTex, 83 pages, 26 figures. Submitted to Review of Modern Physic