Systematic derivation of a generalized<i>t</i>-<i>J</i>model

Abstract

We perform a consistent low-energy reduction of the three-band Hubbard model ${\mathit{H}}_{3\mathit{b}}$ to an effective one-band model ${\mathit{H}}_{1\mathit{b}}$, using as an intermediate step a renormalized effective spin-fermion model ${\mathit{H}}_{\mathrm{sf}}$ in which ${\mathrm{Cu}}^{+}$ and ${\mathrm{Cu}}^{3+}$ configurations are contained as virtual states. We map ${\mathit{PH}}_{\mathrm{sf}}$P into ${\mathit{H}}_{1\mathit{b}}$ using two projectors P onto different Zhang-Rice singlets: (1) orthogonal singlets constructed using O Wanner functions, (2) nonorthogonal singlets. In case 1 (2) the mapping is almost exact (exact as shown before by Zhang) if direct O-O hopping and the ${\mathrm{Cu}}^{3+}$ (${\mathrm{Cu}}^{+}$) configuration can be neglected.