2 Derivation of effective spin models from a three band model for CuO 2 – planes

The derivation of effective spin models describing the low energy magnetic properties of undoped CuO 2 –planes is reinvestigated. Our study aims at a quantitative determination of the parameters of effective spin models from those of a multi–band model and is supposed to be relevant to the analysis of recent improved experimental data on the spin wave spectrum of La 2 CuO 4. Starting from a conventional three–band model we determine the exchange couplings for the nearest and next–nearest neighbor Heisenberg exchange as well as for 4– and 6–spin exchange terms via a direct perturbation expansion up to 12th (14th for the 4–spin term) order with respect to the copper–oxygen hopping t pd. Our results demonstrate that this perturbation expansion does not converge for hopping parameters of the relevant size. Well behaved extrapolations of the couplings are derived, however , in terms of Padé approximants. In order to check the significance of these results from the direct perturbation expansion we employ the Zhang– Rice reformulation of the three band model in terms of hybridizing oxygen Wannier orbitals centered at copper ion sites. In the Wannier notation the perturbation expansion is reorganized by an exact treatment of the strong site–diagonal hybridization. The perturbation expansion with respect to the weak intersite hybridizations is calculated up to 4th order for the Heisenberg coupling and up to 6th order for the 4–spin coupling. It shows excellent convergence and the results are in agreement with the Padé approximants of the direct expansion. The relevance of the 4–spin coupling as the leading correction to the nearest neighbor Heisenberg model is emphasized.