Goldstone modes in Larkin-Ovchinnikov-Fulde-Ferrell superconductors

In nonuniform Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) superconductors, both the gauge symmetry and the continuous translational symmetry of the normal state are spontaneously broken. This leads to additional bosonic excitations, or Goldstone modes, corresponding to the deformations of the order parameter amplitude modulation in real space. We derive general expressions for the energy of the phase and elastic Goldstone modes. As an example, the superfluid density and the elastic modulus of a one-dimensional LOFF superconductor are calculated at low temperatures.