Selfconsistent investigation of multichannel superconductivity in a cuprate model system

The origin of high-temperature superconductivity in cuprates is still an unresolved issue. Among the most likely candidates for mediating the Cooper pair condensate are spin fluctuations and the electron-phonon interaction. While the former have long been proposed to be responsible for various observables in the superconducting state, the latter has recently been shown to produce unconventional gap symmetries when vertex corrections are self-consistently taken into account. Here, we develop multichannel Eliashberg theory to incorporate both pairing mechanisms. Solving self-consistently the full-bandwidth, anisotropic Eliashberg equations for a cuprate model system we find the characteristic $d_{x^2 -y^2}$ symmetry of the superconducting gap and a reasonable order of magnitude for $T_c$ ($50-120$ K). We further find that both mechanisms support an unconventional $d$-wave symmetry of the order parameter, yet the electron-phonon interaction is chiefly responsible for the Cooper pairing and high $T_c$, whereas the spin fluctuations have a suppressing effect on the gap magnitude and critical temperature.