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We investigate the role of electron-electron and electron-phonon interactionsin strongly correlated systems by performing unbiased quantum Monte Carlosimulations in the square lattice Hubbard-Holstein model at half-filling. Westudy the competition and interplay between antiferromagnetism (AFM) andcharge-density wave (CDW), establishing its very rich phase diagram. In theregion between AFM and CDW phases, we have found an enhancement ofsuperconducting pairing correlations, favoring (nonlocal) $s$-wave pairs. Ourstudy sheds light over past inconsistencies in the literature, in particularthe emergence of CDW in the pure Holstein model case.

Phase diagram of the two-dimensional Hubbard-Holstein model: enhancement of $s$-wave pairing between charge and magnetic orders