Indirect RKKY-Type Interaction by Direct Oxygen Hopping

The three-band Emery model, describing the holes in the CuO$\text{}_{2}$ planes of the high-temperature superconducting oxides, is considered. The model includes the direct oxygen-oxygen hopping integral t$\text{}_{pp}$. The exact Bogolyubov transformation is used to exclude one oxygen band and obtain a two-dimensional Anderson model. Afterward, the effective Hamiltonian is obtained by eliminating the second oxygen band with the use of two approximate canonical transformations. The effective Hamiltonian describes the spins residing on the copper sites and interacting through an indirect interaction J$\text{}_{SX}$(R), where R is the distance between two copper ions. J$\text{}_{SX}$(R) depends on the doping rate δ and is a decaying function of R. Numerical results for J$\text{}_{SX}$(R) are given for different doping rates δ for the case of parabolic bands. The obtained interaction J$\text{}_{SX}$(R), when added to the original antiferromagnetic interaction (present in oxides at δ = 0), might lead to a frustration of the long-range antiferromagnetic ordering upon doping.