ANISOTROPIC SPIN HAMILTONIANS DUE TO SPIN-ORBIT AND COULOMB EXCHANGE INTERACTIONS

Here we correct, extend, and clarify results concerning the spin Hamil tonian Hs used to describe the ground manifold of Hubbard models for m agnetic insulators in the presence of spin-orbit interactions. Most of our explicit results are for a tetragonal lattice as applied to some of the copper oxide lamellar systems and are obtained within the appro ximation that H-S consists of a sum of nearest-neighbor bond Hamiltoni ans. We consider both a ''generic'' model in which hopping takes place from one copper ion to another and a ''real'' model in which holes ca n hop from a copper ion to an intervening oxygen 2p band. Both models include orbitally dependent direct and exchange Coulomb interactions i nvolving two orbitals. Our analytic results have been confirmed by num erical diagonalizations for two holes occupying any of the 3d states a nd, if applicable, the oxygen 2p states. An extension of the perturbat ive scheme used by Moriya is used to obtain analytic results for H-S u p to order t(2) (t is the matrix of hopping coefficients) for arbitrar y crystal symmetry for both the ''generic'' and ''real'' models. With only direct orbitally independent Coulomb interactions, our results re duce to Moriya's apart from some minor modifications. For the tetragon al case, we show to all orders in t and lambda, the spin-orbit couplin g constant, that H-S is isotropic in the absence of Coulomb exchange t erms and assuming only nearest-neighbor hopping. In the presence of Co ulomb exchange, scaled by K, the anisotropy in H-S is biaxial and is s hown to be of order Kt(2) lambda(2). Even when K = 0, for systems of s ufficiently low symmetry, the anisotropy in H-S is proportional to t(6 ) lambda(2) when the direct on-site Coulomb interaction U is independe nt of the orbitals involved and of order t(2) lambda(2) otherwise. The se latter results apply to the orthorhombic phase of La2CuO4.