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

Abstract

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 \ensuremath{\varepsilon}), 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.We find that this asymmetry, which vanishes in the extreme Mott-Hubbard regime (${\mathit{U}}_{\mathit{d}}$\ensuremath{\ll}\ensuremath{\varepsilon}), is quite appreciable in the charge-transfer regime (${\mathit{U}}_{\mathit{d}}$\ensuremath{\gtrsim}\ensuremath{\varepsilon}), particularly for the effective Coulomb interactions. We show that for doped holes (forming Zhang-Rice singlets) on neighboring cells the interaction induced by ${\mathit{V}}_{\mathit{pd}}$ can even be attractive due to locally enhanced pd hybridization, while this cannot occur for electrons. The Coulomb interaction induced by ${\mathit{U}}_{\mathit{p}}$ is always repulsive; in addition ${\mathit{U}}_{\mathit{p}}$ gives rise to a ferromagnetic spin-spin interaction which opposes antiferromagnetic superexchange. We show that for hole-doped systems this leads to a subtle cancellation of attractive and repulsive contributions, due to antiferromagnetic and charge-polarization effects, to the net static interaction in a charge-spin (t-J-V) model, and we discuss the significance of this result. The asymmetry in the ee, hh, and eh effective hopping parameters can be particularly large for next-nearest neighbors. Specializing to cuprate parameters, we find that the asymmetry in the nearest-neighbor hopping parameters almost vanishes (accidentally), while the next-nearest-neighbor hopping parameter ${\mathit{t}}^{\ensuremath{'}}$ is close to zero for electrons but is appreciable for holes (${\mathit{t}}^{\ensuremath{'}}$\ensuremath{\approxeq}-0.06 eV). The effective Coulomb interaction between doped holes is found to be repulsive, and even slightly larger than for electrons. All the underlying d-p parameters make significant contributions to the effective interactions and it is shown that certain approximations, such as ${\mathit{U}}_{\mathit{d}}$=\ensuremath{\infty} and ${\mathit{t}}_{\mathit{pp}}$=0, can be qualitatively incorrect. \textcopyright{} 1996 The American Physical Society.