Goldstone modes in Larkin-Ovchinnikov-Fulde-Ferrell superconductors

The order parameter in LOFF superconductors can break translational symmetry, as well as the phase-rotation symmetry, which leads to the existence of additional Goldstone modes. We derive the energy of these modes microscopically, both in the single plane wave (FF) and the two plane wave (LO) phases, and also calculate the superfluid density and the elastic moduli of the nonuniform superconducting phases.