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We study the Holstein-Hubbard model at half filling to explore ordered phasesincluding su- perconductivity (SC), antiferromagnetism (AF), and charge order(CO) in situations where the electron-electron and electron-phonon interactionsare strong (~ electronic bandwidth). The model is solved in the dynamicalmean-field approximation using a continuous-time quantum Monte Carlo impuritysolver. We determine the superconducting transition temperature (Tc) and the SCorder parameter and show that the phonon-induced retardation or the strongCoulomb interaction leads to a significant reduction and shift of the Tc domeif one interprets the system as having an effective static interaction Ueffgiven by the Hubbard U reduced by the phonon-mediated attraction in the staticlimit. This behavior is analyzed by comparison to an effective static model inthe polaron representation with a reduced bandwidth. We also determine thefinite-temperature phase diagram including AF and CO. In the moderate-couplingregime, there is a hysteretic region of AF and CO around Ueff = 0, while thetwo phases are separated by a paramagnetic metal in the weak-coupling regimeand a paramagnetic insulator in the strong coupling regime.

Ordered phases in the Holstein-Hubbard model: Interplay of strong Coulomb interaction and electron-phonon coupling