Superconducting instability in the Holstein-Hubbard model: A numerical renormalization-group study

We have studied the d-wave pairing instability in the two-dimensional Holstein-Hubbard model at the level of a full fluctuation-exchange approximation which treats both Coulomb and electron-phonon (EP) interaction diagrammatically on an equal footing. A generalized numerical renormalization-group technique has been developed to solve the resulting self-consistent field equations. The d-wave superconducting phase diagram shows an optimal Tc at electron concentration 〈n〉~0.9 for the purely electronic Hubbard system. The EP interaction suppresses the d-wave Tc which drops to zero when the phonon-mediated on-site attraction Uₚ becomes comparable to the on-site Coulomb repulsion U. The isotope exponent α is negative in this model and small compared to the classical BCS value αBCS=1/2 or compared to typical observed values in nonoptimally doped cuprate superconductors.