Optical sum rule violation, superfluid weight, and condensation energy in the cuprates

The model of hale superconductivity predicts that the superfluid weight in the zero-frequency delta function in the optical conductivity has an anomal ous contribution from high frequencies, due to lowering of the system's kin etic energy upon entering the superconducting state. The lowering of kineti c energy, mainly in-plane in origin, accounts Tol. both the condensation en ergy of the superconductor as well as an increased potential energy due to larger Coulomb repulsion in the paired state. It leads to an apparent viola tion of the conductivity sum rule, which in the clean limit we predict to b e substantially larger for in-plane than for c-axis conductivity. However, because cuprates are in the dirty limit for c-axis transport, the sum rule violation is found to be greatly enhanced in the c direction. The model pre dicts the sum rule violation to be largest in the underdoped regime and to decrease with doping, more rapidly in the c direction than in the plane. So far, experiments have detected sum rule violation in c-axis transport in s everal cuprates, as well as a decrease and disappearance of this violation for increasing doping, but no violation in plane. We explore the prediction s of the model for a wide range of parameters, both in the absence and in t he presence of disorder, and the relation with current experimental knowled ge.