Phase fluctuations in superconductors: from Galilean invariant to quantum 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 XY model both in the quantum and in the classical regime. We find new dynamic anharmonic vertices, absent in the quantum 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 XY model by a factor at least of order $1/(k_F\xi_0)^2$.