Dynamical mean-field analysis of ordered phases in the half-filled Holstein-Hubbard model

We investigate ordered phases (s-wave superconducting (SC), antiferromagnetic (AF) and charge ordered (CO) phases) of the half-filled Holstein-Hubbard model using the dynamical mean-field theory in combination with a continuous time quantum Monte Carlo impurity solver. We particularly focus on the parameter regime where the electron-electron Coulomb interaction and the phonon mediated retarded attractive interaction are both comparable to the bare electron bandwidth to understand the interplay of these interactions and the retardation effects due to finite phonon frequency \omega_0. Results for the transition temperatures for SC and AF as a function of the effective interaction U_eff(=U-\lambda with U the Coulomb repulsion and \lambda the phonon-mediated attraction) show deviations from the the corresponding results in the anti-adiabatic limit. These are analyzed by comparing with an effective static model in the polaron picture. We also determine the phase diagram around U_eff=0 at low, but nonzero temperatures to show that paramagnetic phases appear between AF and CO when U and \lambda are small or large, while AF and CO directly compete with each other at moderate U and \lambda.