Effective action for phase fluctuations in d-wave superconductors near a Mott transition

Phase fluctuations of a d-wave superconducting order parameter are theoretically studied in the context of high-T<Subscript>c</Subscript> cuprates. We consider an extended t-J model describing electrons in a layer which also contains long-range Coulomb interactions. The constraint of having at most singly occupied sites is enforced by an additional Hubbard term. The Heisenberg interaction is decoupled by a d-wave order parameter in the particle-particle channel. Assuming first that the equilibrium state has long-range phase order, the effective action <InlineEquation ID="Equ1"> <EquationSource Format="TEX">$\mathcal{S}_\textit{eff}$</EquationSource> </InlineEquation> is derived perturbatively for small fluctuations within a path integral formalism, in the presence of the Coulomb and Hubbard interaction terms. In a second step, a more general derivation of <InlineEquation ID="Equ2"> <EquationSource Format="TEX">$\mathcal{S}_\textit{eff}$</EquationSource> </InlineEquation> is performed in terms of a gradient expansion which only assumes that the gradients of the order parameter are small whereas the value of the phase may be large. We show that in the phase-only approximation the resulting <InlineEquation ID="Equ3"> <EquationSource Format="TEX">$\mathcal{S}_\textit{eff}$</EquationSource> </InlineEquation> reduces in leading order in the field gradients to the perturbative one which thus allows to treat also the case without long-range phase order or vortices. Our result generalizes previous expressions for <InlineEquation ID="Equ4"> <EquationSource Format="TEX">$\mathcal{S}_\textit{eff}$</EquationSource> </InlineEquation> to the case of interacting electrons, is explicitly gauge invariant, and avoids problematic singular gauge transformations. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007