Impact of a finite cut-off for the optical sum rule in the superconducting state

A single band optical sum rule derived by Kubo can reveal a novel kind of superconducting state. It relies, however, on a knowledge of the single band contribution from zero to infinite frequency. A number of experiments on the high temperature superconductors over the past five years have used this sum rule; their data has been interpreted in support of “kinetic energy-driven superconductivity.” However, because of the presence of unwanted interband optical spectral weight, they necessarily have to truncate their sum at a finite frequency. This work examines theoretical models where the impact of this truncation can be examined first in the normal state, and then in the superconducting state. The latter case is particularly important as previous considerations attributed the observed anomalous temperature dependence as an artifact of a noninfinite cut-off frequency. We find that this is, in fact, not the case, and that the sign of the corrections from the use of a noninfinite cut-off is such that the observed temperature dependence is even more anomalous when proper account is taken of the cutoff. This conclusion holds if a finite scattering rate is used, which is in contrast to the often considered but less realistic Mattis-Bardeen (dirty) limit. On the other hand, in these same models, we find that the strong observed temperature dependence in the normal state can be attributed to the effect of a noninfinite cut-off frequency.