Optical integral and sum-rule violation in high- T_c superconductors

The purpose of this work is to investigate the role of the lattice in the optical Kubo sum rule in the cuprates. We compute conductivities, optical integrals W, and the change between W in superconducting and normal states (ΔW= W_SC - W_NS) for two-dimensional systems with lattice dispersion typical of the cuprates. We study four different models-a dirty BCS model, a single Einstein boson model, a marginal Fermi-liquid model, and a collective boson model with a feedback from superconductivity on a collective boson. The goal of the paper is twofold. First, we analyze the dependence of W on the upper cutoff (ω_c) placed on the optical integral because in experiments W is measured up to frequencies of order bandwidth. For a BCS model, the Kubo sum rule is almost fully reproduced at ω_c equal to the bandwidth. But for other models only 70-80% of Kubo sum rule is obtained up to this scale and even less so for ΔW, implying that the Kubo sum rule has to be applied with caution. Second, we analyze the sign of ΔW. In all models we studied ΔW is positive at small ω_c, then crosses zero and approaches a negative value at large ω_c, i.e., the optical integral in a superconductor is smaller than in a normal state within the one band model. The point of zero crossing, however, increases with the interaction strength and in a collective boson model becomes comparable to the bandwidth at strong coupling. We argue that this model exhibits the behavior consistent with that in the cuprates.