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The derivation of the effective theory for the phase degrees of freedom in asuperconductor is still, to some extent, an open issue. It is commonly assumedthat the classical XY model and its quantum generalizations can be exploited aseffective phase-only models. In the quantum regime, however, this assumptionleads to spurious results, such as the violation of the Galilean invariance inthe continuum model. Starting from a general microscopic model, in this paperwe explicitly derive the effective low-energy theory for the phase, up tofourth-order terms. This expansion allows us to properly take into accountdynamic effects beyond the Gaussian level, both in the continuum and in thelattice model. After evaluating the one-loop correction to the superfluiddensity we critically discuss the qualitative and quantitative differencesbetween the results obtained within the quantum XY model and within the correctlow-energy theory, both in the case of s-wave and d-wave symmetry of thesuperconducting order parameter. Specifically, we find dynamic anharmonicvertices, which are absent in the quantum XY model, and are crucial to restoreGalilean invariance in the continuum model. As far as the more realisticlattice model is concerned, in the weak-to-intermediate-coupling regime we findthat the phase-fluctuation effects are quantitatively reduced with respect tothe XY model. On the other hand, in the strong-coupling regime we show that thecorrespondence between the microscopically derived action and the quantum XYmodel is recovered, except for the low-density regime.Comment: 29 pages, 11 figures. Slightly revised presentation, accepted for  publication in Phys. Rev.

physical review. b, condensed matter and materials physics

Low-energy phase-only action in a superconductor: A comparison with the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>XY</mml:mi></mml:math>model

Low-energy phase-only action in a superconductor: A comparison with theXYmodel