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By using variational wave functions and quantum Monte Carlo techniques, weinvestigate the interplay between electron-electron and electron-phononinteractions in the two-dimensional Hubbard-Holstein model. Here, theground-state phase diagram is triggered by several energy scales, i.e., theelectron hopping $t$, the on-site electron-electron interaction $U$, the phononenergy $\omega_0$, and the electron-phonon coupling $g$. At half filling, theground state is an antiferromagnetic insulator for $U \gtrsim 2g^2/\omega_0$,while it is a charge-density-wave (or bi-polaronic) insulator for $U \lesssim2g^2/\omega_0$. In addition to these phases, we find a superconducting phasethat intrudes between them. For $\omega_0/t=1$, superconductivity emerges whenboth $U/t$ and $2g^2/t\omega_0$ are small; then, by increasing the value of thephonon energy $\omega_0$, it extends along the transition line betweenantiferromagnetic and charge-density-wave insulators. Away from half filling,phase separation occurs when doping the charge-density-wave insulator, while auniform (superconducting) ground state is found when doping the superconductingphase. In the analysis of finite-size effects, it is extremely important toaverage over twisted boundary conditions, especially in the weak-coupling limitand in the doped case.

Superconductivity, charge-density waves, antiferromagnetism, and phase separation in the Hubbard-Holstein model