Phase fluctuations in superconductors: From Galilean invariant to quantum $\mathrm{XY}$ models

We analyze the corrections to the superfluid density due to phase fluctuations within both a continuum and a lattice model for s- and d-wave superconductors. We expand the phase-only action beyond the Gaussian level and compare our results with the quantum $\mathrm{XY}$ model both in the quantum and in the classical regime. We find dynamic anharmonic vertices, absent in the quantum $\mathrm{XY}$ model, which are responsible for the vanishing of the correction to the superfluid density at zero temperature in a continuum (Galilean invariant) model. Moreover, the phase-fluctuation effects are reduced with respect to the $\mathrm{XY}$ model by a factor at least of order ${1/(k}_F{\xi }_0{)}^2.$