Inhomogeneous superconductivity in condensed matter and QCD

Inhomogeneous superconductivity arises when the species participating in the pairing phenomenon have different Fermi surfaces with a large enough separation. In these conditions it could be more favorable for each of the pairing fermions to stay close to its Fermi surface and, unlike the usual BCS state, for the Cooper pair to have a nonzero total momentum. For this reason, in this state the gap varies in space, the ground state is inhomogeneous, and a crystalline structure might be formed. This situation was considered for the first time by Fulde and Ferrell (1964) and Larkin and Ovchinnikov (1964), after whom the corresponding state is called the LOFF state. The spontaneous breaking of the space symmetries in the vacuum state is a characteristic feature of this phase and is associated with the presence of long-wavelength excitations of zero mass. The situation described here is of interest both in solid-state and in elementary-particle physics, in particular in quantum chromodynamics at high density and low temperature. This review presents the theoretical approach to the LOFF state and its phenomenological applications using the language of the effective field theories.