Effects of surface exchange anisotropy in Heisenberg ferromagnetic insulators

We consider an fcc semi-infinite ferromagnetic insulator displaying an anisotropic exchange interaction between spins on the (111) surface plane of the form $J_{\parallel }(S_i^x{S}_j^x+S_i^y{S}_J^y+\eta S_i^z{S}_j^z)$. We assume all other interactions isotropic. A self-consistent random-phase-approximation calculation is performed, with a Green's-function method valid for any spin $S$, up to the bulk transition temperature $T_c^b$, by the assumption that the magnetization of the third layer equals the bulk value. For $\eta$ sufficiently large, the surface magnetization is nonzero for $T>\mathrm{}{T}_c^b$, up to a transition temperature $T_c^s\left(\eta \right)$ whenever $\eta \ge {\eta }_c>1\mathrm{}$, where $T_c^s\left({\eta }_c\right)=T_c^b$. For $T>\mathrm{}{T}_c^b$ the system is equivalent to a film of three layers, where the magnetization of the third one is identically zero as a boundary condition. A discontinuity of the derivative in the curve of the magnetization of the first two layers versus temperature is found at $T_c^b$. The results show clearly a crossover from Heisenberg to Ising behavior at the surface.