Superexchange anisotropy in the cuprates

The single-bond superexchange anisotropy resulting from the 2pσ and the 2pz orbitals on the ligand ion of the cuprates is analyzed for a general case in which the oxygen octahedra surrounding each copper rotate around a general axis in the CuO2 plane. An extension of Moriya's calculation shows that when the oxygen orbitals are degenerate, the single-bond spin interaction is rotationally invariant, contrary to previous claims. When the orbitals are nondegenerate the single-bond Hamiltonian is anisotropic, but the anisotropy is small. The ground state of the entire CuO2 plane is found to be completely antiferromagnetic in the low-temperature tetragonal phase. The weak ferromagnetism of the low-temperature orthorhombic phase is mainly due to the anisotropy resulting from frustration of the single-bond interactions. These frustration effects also generate an anisotropy which dominates that due to the nondegeneracy of the oxygen orbitals. We present explicit results for the weak ferromagnetic moment and for the spin-wave spectrum, as a function of the direction of the octahedra rotation axis.