Electromagnetic fluctuation and collective modes in relativistic bosonic superfluid in mixed dimensions

In Gaussian approximation, we investigate the marginal electromagnetic fluctuation in models of charged relativistic bosonic superfluids in three and two spatial dimensions at zero temperature. The electromagnetism is modeled by the ordinary Maxwell term and the non-local pseudo-electrodynamics action in these dimensions respectively. We explore the collective excitations in these systems by integrating the superfluid velocity fields. In (3+1) dimensions, we derive the roton mode reminiscent of what was discovered in the context of the free relativistic Bose-Einstein condensate as a generalization of the Higgs mode and determine the necessary and sufficient condition for the roton to exist. In (2+1) dimensions, besides solving the dispersion relation for the surface plasmon, we prove there cannot be roton-like excitation in this model as opposed to its (3+1) dimensional counterpart, and additionally derive the asymptotic lines of the dispersion in the limits of long wavelength and short distance.