Temperature dependence of the single-ion lattice anisotropy coefficients of a ferromagnetic Heisenberg monolayer

In investigations of the orientation of the magnetization of thin ferromagnetic films, the single-ion anisotropy coefficients play a crucial role. By applying a thermodynamic perturbation theory we calculate the temperature dependence of the second- and fourth-order single-ion anisotropies for a Heisenberg monolayer with Tyablikov decoupling (the random-phase approximation, RPA) and compare with results obtained from mean-field theory. In order to assess the accuracy of the Tyablikov (RPA) and also the Callen decoupling approximations in the Green's function many-body theory, we calculate the magnetization of a Heisenberg spin pair and of a ferromagnetic monolayer, and compare the results with exact solutions available for the spin pair with arbitrary spins, and a recent quantum Monte Carlo calculation (for spin 1/2) for the monolayer. The RPA decoupling provides a fairly good approximation to the exact results for the magnetization over the whole temperature range of interest. Because of this, we expect the calculated anisotropy coefficients to be approximated well by this method.