Many-body Green's function theory for thin ferromagnetic films: exact   treatment of the single-ion anisotropy
  A theory for the magnetization of ferromagnetic films is formulated within the framework of many-body Green's funtion theory which considers all components of the magnetization. The model Hamiltonian includes a Heisenberg term, an external field, a second- and fourth-order uniaxial single-ion anisotropy, and the magnetic dipole-dipole coupling. The single-ion anisotropy terms can be treated exactly by introducing higher-order Green's functions and subsequently taking advantage of relations between products of spin operators which leads to an automatic closure of the hierarchy of the equations of motion for the Green's functions with respect to the anisotropy terms. This is an improvement on the method of our previous work, which treated the corresponding terms only approximately by decoupling them at the level of the lowest-order Green's functions. RPA-like approximations are used to decouple the exchange terms in both the low-order and higher-order Green's functions. As a first numerical example we apply the theory to a monolayer for spin S=1 in order to demonstrate the superiority of the present treatment of the anisotropy terms over the previous approximate decouplings. 
P. Froebrich, P.J. Kuntz, M. Saber