Phase fluctuations of a d-wave superconducting order parameter aretheoretically studied in the context of high-T$_c$ cuprates. We consider the$t-J$ model describing layered compounds, where the Heisenberg interaction isdecoupled by a d-wave order parameter in the particle-particle channel.Assuming first that the equilibirum state has long-range phase order, theeffective action $\mathcal{S}_{eff}$ is derived perturbatively for smallfluctuations within a path integral formalism, in the presence of the Coulomband Hubbard interaction terms. In a second step, a more general derivation of$\mathcal{S}_{eff}$ is performed in terms of a gradient expansion which onlyassumes that the gradients of the order parameter are small whereas the valueof the phase may be large. We show that in the phase-only approximation theresulting $\mathcal{S}_{eff}$ reduces in leading order in the field gradientsto the perturbative one which thus allows to treat also the case withoutlong-range phase order or vortices. Our result generalizes previous expressionsfor $\mathcal{S}_{eff}$ to the case of interacting electrons, is explicitlygauge invariant, and avoids problematic singular gauge transformations.
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