We consider the evolution of d-wave pairing, mediated by nearly critical spin fluctuations, with the coupling strength. We show that the onset temperature for pairing, T*, smoothly evolves between weak and strong coupling, passing through a broad max
imum at intermediate coupling. At strong coupling, T* is of order the magnetic exchange energy J. We argue that for all couplings, pairing is confined to the vicinity of the Fermi surface. We also find that thermal spin fluctuations only modestly reduce T*, even at criticality, but they substantially smooth the gap anisotropy. The latter evolves with coupling, being the largest at weak coupling.
Much attention has been given to a possible violation of the optical sum rule in the cuprates, and the connection this might have to kinetic energy lowering. The optical integral is composed of a cut-off independent term (whose temperature dependence
is a measure of the sum rule violation), plus a cut-off dependent term that accounts for the extension of the Drude peak beyond the upper bound of the integral. We find that the temperature dependence of the optical integral in the normal state of the cuprates can be accounted for solely by the latter term, implying that the dominant contribution to the observed sum rule `violation in the normal state is due to the finite cut-off. This cut-off dependent term is well modeled by a theory of electrons interacting with a broad spectrum of bosons.
