DERIVATION AND NUMERICAL STUDY OF THE SINGLET-TRIPLET MODEL FOR CUPRATE SUPERCONDUCTORS

We perform a low-energy reduction of the three-band Hubbard Hamiltonian (H-3b), 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 x(2)-y(2) symmetry. We solve exactly the resulting Hamiltonian H-T in a system of 2 x 2 unit cells. From the analytical dependence of the parameters of H-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 Delta, very large Cu on-site Coulomb repulsion U-d, and O-O hopping t(pp) = 0. This fact is in contrast with the mapping of H-3b to a one-band model using nonorthogonal singlets, which is very accurate when the Cu+ configuration can be neglected. Although the amount of local triplet states in the low-energy eigenstates is in general small, it increases with t(pp) and for large t(pp) 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.