We present a detailed analysis of the magnetic, density, and superconducting fluctuations of the two-dimensional Hubbard-Holstein model with a dispersionless Einstein phonon as a paradigmatic model for electron-electron and electron-phonon interactions. Using the functional renormalization group in the single-boson exchange formulation, we were able to resolve the full frequency dependence of the two-particle vertex and take into account the feedback from the electronic self-energy. We provide a systematic analysis of the leading magnetic, density, and superconducting susceptibilities ranging from the adiabatic to the anti-adiabatic regime in order to investigate the rich physical picture emerging from intertwined ordering tendencies. In particular, we assess the importance of self-energy effects leading to an enhancement of the d-wave superconducting susceptibility for increasing phonon frequencies. In contrast, at low frequencies, large density contributions eventually induce a reduction of s-wave superconductivity with increasing electron-phonon coupling, indicating the breakdown of Migdal-Eliashberg theory.

[1] https://journals.aps.org/prresearch/abstract/10.1103/6vyb-h379