Doping dependence and multichannel mediators of superconductivity: Calculations for a cuprate model

We study two aspects of the superconductivity in a cuprate model system, its doping dependence and the influence of competing pairing mediators. We first include electron-phonon interactions beyond Migdal's approximation and solve self-consistently, as a function of doping and for an isotropic electron-phonon coupling, the full-bandwidth, anisotropic vertex-corrected Eliashberg equations under a non-interacting state approximation for the vertex correction. Our results show that such pairing interaction supports the experimentally observed $d_{x^2-y^2}$-wave symmetry of the superconducting gap, but only in a narrow doping interval of the hole-doped system. Depending on the coupling strength, we obtain realistic values for the gap magnitude and superconducting critical temperature $T_c$ close to optimal doping, rendering the electron-phonon mechanism an important candidate for mediating superconductivity in this model system. Second, for a doping near optimal hole doping, we study multichannel superconductivity, by including both vertex-corrected electron-phonon interaction and spin and charge fluctuations as pairing mechanisms. We find that both mechanisms cooperate to support an unconventional $d$-wave symmetry of the order parameter, yet the electron-phonon interaction is mainly responsible for the Cooper pairing and high critical temperature $T_c$. Spin fluctuations are found to have a suppressing effect on the gap magnitude and critical temperature due to their repulsive interaction at small coupling wave vectors.