Intertwined fluctuations and isotope effects in the Hubbard-Holstein model on the square lattice from functional renormalization

Electron-electron and electron-phonon interactions are responsible for the formation of spin, charge, and superconducting correlations in layered quantum materials. A paradigmatic model for such materials that captures both kinds of interactions is the two-dimensional Hubbard-Holstein model with a dispersionless Einstein phonon. In this work, we provide a detailed analysis of the magnetic, density, and superconducting fluctuations at and away from half-filling. To that end, we employ the functional renormalization group using the recently introduced extension of the single-boson exchange formulation. More precisely, we go beyond previous approaches to the model by resolving the full frequency dependence of the two-particle vertex and taking into account the feedback from the electronic self-energy. We perform broad parameter scans in the space of Hubbard repulsion, electron-phonon coupling strength, and phonon frequency to explore the leading magnetic, density, and superconducting susceptibilities from the adiabatic to the anti-adiabatic regime. Our numerical data reveal that self-energy effects lead to an enhancement of the $d$-wave superconducting susceptibility towards larger phonon frequencies, in contrast to earlier isotope-effect studies. At small phonon frequencies, large density contributions to the $s$-wave superconducting susceptibility change sign and eventually lead to a reduction of $s$-wave superconductivity with increasing electron-phonon coupling, signaling the breakdown of Migdal-Eliashberg theory. We analyze our findings systematically, employing detailed diagnostics of the intertwined fluctuations and pinning down the various positive and negative isotope effects of the physical susceptibilities.