Competition between antiferromagnetic and charge density wave order in   the half filled Hubbard-Holstein model
  We present a determinant quantum Monte Carlo study of the competition between instantaneous on-site Coulomb repulsion and retarded phonon-mediated attraction between 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 of AFM-CDW competition that incorporates the phonon mediated attraction into an effective-U Hubbard model requires significant refinement. Specifically, retardation effects slow the onset of charge order, so that CDW order remains absent even when the effective U is negative. This delay opens a window where neither AFM nor CDW order is well established, and where there are signatures of a possible metallic phase. 
E. A. Nowadnick, S. Johnston, B. Moritz, R. T. Scalettar, T. P.   Devereaux
cond-mat.str-el