Derivation and numerical study of the singlet-triplet model for cuprate superconductors

We perform a low-energy reduction of the three-band Hubbard Hamiltonian (${\mathit{H}}_{3\mathit{b}}$), keeping in the relevant Hilbert subspace not only local singlets (Zhang-Rice singlets), but also triplet states between Cu holes and O holes at the Wannier function of the same site, with ${\mathit{x}}^2$-${\mathit{y}}^2$ symmetry. We solve exactly the resulting Hamiltonian ${\mathit{H}}_{\mathit{T}}$ in a system of 2×2 unit cells. From the analytical dependence of the parameters of ${\mathit{H}}_{\mathit{T}}$ and the numerical results, one can see that the local triplet states can be practically neglected for finite O-Cu on-site energy difference Δ, very large Cu on-site Coulomb repulsion ${\mathit{U}}_{\mathit{d}}$, and O-O hopping ${\mathit{t}}_{\mathit{p}\mathit{p}}$=0. This fact is in contrast with the mapping of ${\mathit{H}}_{3\mathit{b}}$ to a one-band model using nonorthogonal singlets, which is very accurate when the ${\mathrm{Cu}}^{+}$ configuration can be neglected. Although the amount of local triplet states in the low-energy eigenstates is in general small, it increases with ${\mathit{t}}_{\mathit{p}\mathit{p}}$ and for large ${\mathit{t}}_{\mathit{p}\mathit{p}}$ it is necessary to introduce higher-order corrections in the one-band model to accurately represent the low-energy physics. In all cases even when local triplets are not important, the t-J model should be supplemented with other terms, to describe the lowest-energy levels. We also discuss briefly the effect of nonbonding O orbitals.