Collective modes in Fulde-Ferrell-Larkin-Ovchinnikov superconductors: Role of long-range Coulomb interaction and signatures in density response

We theoretically investigate collective excitations in the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states of Pauli-limited superconducting films. When the long-range Coulomb interaction is absent, excitation spectra consist of two gapless and three gapped modes. The gapless modes are the Nambu-Goldstone modes associated with the spontaneous breaking of the $\mathrm{U}(1)$ symmetry and the translational symmetry. The gapped modes include the Higgs mode and the twofold degenerate modes that cause the oscillation of the domain width and grayness of FFLO nodal planes. We find that the long-range Coulomb interaction only gaps out the gapless phase mode through the Anderson-Higgs mechanism, while the other modes remain unaffected. Furthermore, the field evolution of the dispersion of the gapless elastic mode, the Nambu-Goldstone mode associated with the translational symmetry breaking, is associated with that of the bandwidth of the mid-gap Andreev bound states. We demonstrate that the signature of the elastic mode can be detected by measuring the density-density response function.