Dynamical Mean-Field Theory of Superconductivity in the Hubbard-Holstein Model

We present a study of the half-filled Hubbard-Holstein model with superconducting order parameter employing the dynamical mean-field theory in combination with the numerical renormalization group. The Hubbard-Holstein model is a prototype model for understanding the interplay between the local Coulomb repulsion U and the electron-phonon coupling g. The ground state is metallic when both U and g are small, but is a Mott-Hubbard insulator (MHI) when U is larger than the critical value of the Coulomb repulsion U_c and a bipolaron insulator (BPI) when g is larger than the critical value of the electron-phonon coupling g_c. Here, we investigate the interplay between the electron-electron and electron-phonon interactions in superconducting state emerging around the phase boundary between metallic and insulating states. In particular, the effect of phonon softening on supercondcutivity is investigated in the strong correlation regime. The variation of the superconductivity is also probed by changing the phonon frequency from adiabatic to nonadiabatic regimes.