Limitations Of Constrained Random Phase Approximation Downfolding

We check the accuracy of the constrained random phase approximation (cRPA) downfolding scheme by considering one-dimensional two- and three-orbital Hubbard models with a target band at the Fermi level and one or two screening bands away from the Fermi level. Using numerically exact quantum Monte Carlo simulations of the full and downfolded model, we demonstrate that depending on filling, the effective interaction in the low-energy theory is either barely screened or antiscreened, in contrast to the cRPA prediction. This observation is explained by a functional renormalization group analysis, which shows that the cRPA contribution to the screening is to a large extent canceled by other diagrams in the direct particle-hole channel. We comment on the implications of this finding for the ab initio estimation of interaction parameters in low-energy descriptions of solids.