By using variational wave functions and quantum Monte Carlo techniques, we investigate the interplay between electron-electron and electron-phonon interactions in the two-dimensional Hubbard-Holstein model. Here, the ground-state phase diagram is triggered by several energy scales, i.e., the electron hopping t, the on-site electron-electron interaction U, the phonon energy ω₀, and the electron-phonon coupling g. At half filling, the ground state is an antiferromagnetic insulator for U2g²/ω₀, while it is a charge-density-wave (or bipolaronic) insulator for U2g²/ω₀. In addition to these phases, we find a superconducting phase that intrudes between them. For ω₀/t=1, superconductivity emerges when both $U/t$ and 2g²/tω₀ are small; then, by increasing the value of the phonon energy ω₀, it extends along the transition line between antiferromagnetic and charge-density-wave insulators. Away from half filling, phase separation occurs when doping the charge-density-wave insulator, while a uniform (superconducting) ground state is found when doping the superconducting phase. In the analysis of finite-size effects, it is extremely important to average over twisted boundary conditions, especially in the weak-coupling limit and in the doped case.
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Karakuzu et al. (2017) studied this question.