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The effective interaction of downfolded low-energy models for electrons insolids can be obtained by integrating out the high energy bands away from thetarget band near the Fermi level. Here, we apply the constrained random-phaseapproximation (cRPA) and constrained functional renormalization group (cfRG),which can go beyond cRPA by including all one-loop diagrams, to calculate andcompare the effective interactions of the three-band Emery model, which isoften used to investigate cuprate high-temperature superconductors. At halfband filling, we find that the effective interaction increases as the chargetransfer energy ($\Delta_{dp}$) increases and similar behavior is obtained as afunction of the interatomic 2$p$-3$d$ interaction ($U_{dp}$). However, theeffective interaction is more sensitive to $\Delta_{dp}$ than $ U_{dp}$. Formost of the parameter sets, the effective static interaction is overscreened incRPA compared to cfRG. The low-energy models at half-filling are solved withindynamical mean-field theory (DMFT). The results show that despite the differentstatic interactions, the systems with cRPA and cfRG interaction exhibit a Motttransition at similar values of $\Delta_{dp}$. We also investigate theeffective interaction as a function of doping. The cfRG effective interactiondecreases as the electron number increases and displays a trend opposite tothat of cRPA. Antiscreening is observed for the hole-doped case. For all thecases studied, the near-cancellation of the direct particle-hole channel isobserved. This indicates that at least for the downfolding of the onsiteinteraction terms, methods beyond cRPA may be required.

physical review. b./physical review. b

Investigation of the effective interactions for the Emery model by the  constrained random-phase approximation and constrained functional  renormalization group