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 ${\mathrm{Cu}}_4$ ${\mathrm{O}}_8$ cluster, with those obtained with an effective Hamiltonian ${\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}$ in which the Cu-O hopping ${\mathit{t}}_{\mathit{p}\mathit{d}}$ is eliminated by means of a canonical transformation. This transformation up to order ${\mathit{t}}_{\mathit{p}\mathit{d}}^4$ is not quantitatively valid when the relevant energy differences are of the order of √8 ${\mathit{t}}_{\mathit{p}\mathit{d}}$ or less. However, an ${\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}$ with the form of the second-order one, but which retains all higher-order terms contained in a ${\mathrm{CuO}}_4$ (${\mathrm{Cu}}_2$ ${\mathrm{O}}_7$) 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 ${\mathit{H}}_{\mathit{e}\mathit{f}\mathit{f}}$ makes possible the numerical study of the three-band Hubbard model for realistic parameters, using larger clusters.