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

Abstract

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 ${T}_{c}$ at electron concentration $〈n〉\ensuremath{\sim}0.9$ for the purely electronic Hubbard system. The EP interaction suppresses the $d$-wave ${T}_{c}$ which drops to zero when the phonon-mediated on-site attraction ${U}_{p}$ becomes comparable to the on-site Coulomb repulsion $U$. The isotope exponent $\ensuremath{\alpha}$ is negative in this model and small compared to the classical BCS value ${\ensuremath{\alpha}}_{\mathrm{BCS}}=\frac{1}{2}$ or compared to typical observed values in nonoptimally doped cuprate superconductors.