Effective single-band models for the high-Tccuprates. I. Coulomb interactions

Starting with the three-band extended Hubbard model (or d-p model) widely used to represent the ${\mathrm{CuO}}_2$ planes in the high-${\mathit{T}}_{\mathit{c}}$ cuprates, we make a systematic reduction to an effective single-band model using a previously developed cell-perturbation method. The range of parameters for which this mapping is a good approximation is explored in the full Zaanen-Sawatzky-Allen diagram (copper Coulomb repulsion ${\mathit{U}}_{\mathit{d}}$ versus charge-transfer energy ɛ), together with an investigation of the validity of a further mapping to an effective charge-spin (t-J-V) model. The variation of the effective single-band parameters with the parameters of the underlying multi-band model is investigated in detail, and the parameter regime where the model represents the high-${\mathit{T}}_{\mathit{c}}$ cuprates is examined for specific features that might distinguish it from the general case. In particular, we consider the effect of Coulomb repulsions on oxygen (${\mathit{U}}_{\mathit{p}}$) and between copper and oxygen (${\mathit{V}}_{\mathit{pd}}$). We find that the reduction to an effective single-band model is generally valid for describing the low-energy physics, and that ${\mathit{V}}_{\mathit{pd}}$ and ${\mathit{U}}_{\mathit{p}}$ (unless unrealistically large) actually slightly improve the convergence of the cell-perturbation method. Unlike in the usual single-band Hubbard model, the effective intercell hopping and Coulomb interactions are different for electrons and holes.