Effective Hamiltonian for cuprate superconductors

We compare numerical results of O and Cu occupations and optical and magnetic excitations for the three-band Hubbard model in a Cu₄O₈ cluster, with those obtained with an effective Hamiltonian H_eff in which the Cu-O hopping t_pd is eliminated by means of a canonical transformation. This transformation up to order t_pd⁴ is not quantitatively valid when the relevant energy differences are of the order of √8 t_pd or less. However, an H_eff with the form of the second-order one, but which retains all higher-order terms contained in a CuO₄ (Cu₂O₇) cluster for zero or one added hole (one added electron) to the undoped system, leads to a very good agreement with the results using the full Hilbert space. This H_eff makes possible the numerical study of the three-band Hubbard model for realistic parameters, using larger clusters.