Entire set of dynamical excitations of a one-dimensional Fulde-Ferrell-pairing Fermi superfluid based on momentum excitation

We theoretically investigate a one-dimensional Fulde-Ferrell Fermi superfluid at a finite effective Zeeman field $h$ and study all of the dynamical excitations related to density perturbation. By calculating the density dynamic structure factor, we find anisotropic dynamical excitations in both collective modes and single-particle excitations. Along the direction of center-of-mass momentum $p$, there are two obvious gapless collective modes with different speeds and two kinds of continuous regimes related to single-particle excitation. The lower collective modes are from the usual gauge symmetry breaking and have a larger speed than the one in the negative direction of $p$. The higher one is due to the direction spontaneous symmetry breaking of center-of-mass momentum $p$ and separates a gapless lower-branch single-particle excitation from the other gapped higher-branch single-particle excitation in the positive $p$ direction. However, this higher mode disappears in the opposite direction of $p$, where two single-particle excitations overlap with each other. These signals of dynamical excitations could help to distinguish a Fulde-Ferrell superfluid from the conventional Bardeen-Cooper-Schrieffer superfluid in future experiments.