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We present a determinant quantum Monte Carlo study of the competition betweeninstantaneous on-site Coulomb repulsion and retarded phonon-mediated attractionbetween electrons, as described by the two dimensional Hubbard-Holstein model.At half filling, we find a strong competition between antiferromagnetism (AFM)and charge density wave (CDW) order. We demonstrate that a simple picture ofAFM-CDW competition that incorporates the phonon mediated attraction into aneffective-U Hubbard model requires significant refinement. Specifically,retardation effects slow the onset of charge order, so that CDW order remainsabsent even when the effective U is negative. This delay opens a window whereneither AFM nor CDW order is well established, and where there are signaturesof a possible metallic phase.

Competition between antiferromagnetic and charge density wave order in the half filled Hubbard-Holstein model