Effective actions and phase fluctuations in d-wave superconductors

We study effective actions for order-parameter fluctuations at low temperature in layered d-wave superconductors such as the cuprates. The order parameter lives on the bonds of a square lattice and has two amplitude and two phase modes associated with it. The low-frequency spectral weights for amplitude and relative phase fluctuations is determined and found to be subdominant to quasiparticle contributions. The Goldstone phase mode and its coupling to density fluctuations in charged systems is treated in a gauge-invariant manner. The Gaussian phase action is used to study both the c-axis Josephson plasmon and the more conventional in-plane plasmon in the cuprates. We go beyond the Gaussian theory by deriving a coarse-grained quantum $\mathrm{XY}$ model, which incorporates important cutoff effects overlooked in previous studies. A variational analysis of this effective model shows that in the cuprates, quantum effects of phase fluctuations are important in reducing the zero-temperature superfluid stiffness, but thermal effects are small for $T\ll T_c.$