Low-energy phase-only action in a superconductor: A comparison with the $\mathrm{XY}$ model

The derivation of the effective theory for the phase degrees of freedom in a superconductor is still, to some extent, an open issue. It is commonly assumed that the classical $\mathrm{XY}$ model and its quantum generalizations can be exploited as effective phase-only models. In the quantum regime, however, this assumption leads to spurious results, such as the violation of Galilean invariance in the continuum model. Starting from a general microscopic model, in this paper we explicitly derive the effective low-energy theory for the phase, up to fourth-order terms. This expansion allows us to properly take into account dynamic effects beyond the Gaussian level, both in the continuum and in the lattice model. After evaluating the one-loop corrections to the superfluid stiffness we critically discuss the qualitative and quantitative differences between the results obtained within the quantum $\mathrm{XY}$ model and within the correct low-energy theory, both in the case of s- and d-wave symmetry of the superconducting order parameter. Specifically, we find dynamic anharmonic vertices, which are absent in the quantum $\mathrm{XY}$ model, and are crucial to restore Galilean invariance in the continuum model. As far as the more realistic lattice model is concerned, in the weak-to-intermediate-coupling regime we find that the phase-fluctuation effects are quantitatively reduced with respect to the $\mathrm{XY}$ model. On the other hand, in the strong-coupling regime we show that the correspondence between the microscopically derived action and the quantum $\mathrm{XY}$ model is recovered, except for the low-density regime.