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

TL;DR

This paper derives a gauge-invariant effective action for phase fluctuations in d-wave superconductors near a Mott transition, applicable to both ordered and disordered phases, including vortex effects.

Contribution

It generalizes previous effective actions to interacting electrons, incorporating Coulomb and Hubbard interactions, and removes the need for singular gauge transformations.

Findings

Derived a perturbative effective action for small phase fluctuations.

Extended the effective action to include large phase variations and vortices.

Ensured gauge invariance and applicability to disordered phases.

Abstract

Phase fluctuations of a d-wave superconducting order parameter are theoretically studied in the context of high-T$_c$ cuprates. We consider the $t-J$ model describing layered compounds, where the Heisenberg interaction is decoupled by a d-wave order parameter in the particle-particle channel. Assuming first that the equilibirum state has long-range phase order, the effective action $\mathcal{S}_{eff}$ 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 $\mathcal{S}_{eff}$ 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 $\mathcal{S}_{eff}$ 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 $\mathcal{S}_{eff}$ to the case of interacting electrons, is explicitly gauge invariant, and avoids problematic singular gauge transformations.