Symmetry, spin-orbit interactions, and spin anisotropies

The origins of anisotropy in the effective spin Hamiltonian, describing the ground manifold of Hubbard models with spin-orbit interactions, are critically discussed. For tetragonal symmetry, we show exactly that spin anisotropy can arise only if one includes both spin-orbit and Coulomb exchange interactions. For lower symmetries, additional anisotropies arise from terms which were hitherto neglected. Our analytic results are supported by numerical solutions for single bond clusters. These results can explain the easy plane anisotropy in the antiferromagnetic cuprates.