Correlated Metallic Phases in the Hubbard-Holstein Hamiltonians

The one-dimensional Hubbard-Holstein Hamiltonian is rich with phases, with competition between the Hubbard interaction and the density coupling to the Holstein phonon driving a novel correlated metallic phase (the Luther-Emery liquid). We utilize the functional Renormalization Group (fRG) to explore the behavior of the metallic phase in two and three dimensional variants of the model. We construct the flow of both the electron-electron and electron-phonon vertices, with the latter enabling the calculation of phonon softening, the broadening of phonon linewidths, and the onset of charge order. We find a shrunken metallic phase in the three-dimensional model with rich superconducting correlations which show senstivity to the frequency of the Holstein phonon. Finally, we constuct a preliminary phase diagrams of the three-dimensional Hubbard-Holstein model.