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

The model of hole superconductivity predicts that the superfluid weight in the zero-frequency $\delta$ function in the optical conductivity has an anomalous contribution from high frequencies, due to lowering of the system’s kinetic energy upon entering the superconducting state. The lowering of kinetic energy, mainly in-plane in origin, accounts for both the condensation energy of the superconductor as well as an increased potential energy due to larger Coulomb repulsion in the paired state. It leads to an apparent violation of the conductivity sum rule, which in the clean limit we predict to be 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 predicts 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 several cuprates, as well as a decrease and disappearance of this violation for increasing doping, but no violation in plane. We explore the predictions of the model for a wide range of parameters, both in the absence and in the presence of disorder, and the relation with current experimental knowledge.