J. Stein: Nonlocal Coulomb interactions and the magnetic anisotropies in the low-temperature phases of the cuprates, Phys. Rev. B 53 (1996) 785 (Postscript-File)
We have extended and generalized earlier studies of the magnetic anisotropies of the Cu spins in the copper-oxide planes generated by the spin-orbit coupling on the coppers and by the nonlocal Coulomb interactions between neighboring ions. Our derivation of the anisotropic magnetic Hamiltonian includes the contributions from general matrix elements of the Coulomb interaction both between the copper ions of a single bond and between the adjacent copper and oxygen sites. The results apply for a general rotation axis of the oxygen octahedra, with the assumption of tetragonal site symmetry of the copper. This mechanism leads to a form of the anisotropic spin interaction between the Cu ions which is identical to the one generated by kinetic superexchange in the presence of purely local Coulomb interactions. Together with the numerical estimates for the resulting spin couplings, this indicates that the Coulomb direct exchange leads to significant contributions to the isotropic spin interaction and the out-of-plane anisotropy, but does not affect qualitatively the magnetic order in the ground state.