Effects of surface exchange anisotropies on magnetic critical and multicritical behavior at surfaces

The critical behavior of a semi-infinite $n$-vector model with anisotropic ferromagnetic pair interactions on the surface and isotropic interactions everywhere else is studied with the use of renormalization-group methods for dimension $d=4-\epsilon$. The effects of surface anisotropies are analyzed near the isotropic ordinary transition (where they produce corrections to scaling), near the isotropic special transition (where they are relevant perturbations), and near a new class of "anisotropic special" transitions. The latter occur if the surface exchange constants associated with $m_e$ (easy-magnetization) components take a critical value above which $m_e$-component surface order is possible while the interaction constants of the remaining $m_h=n-m_e$ (hard-magnetization) components are weaker. Of particular interest is the anisotropic special transition with an easy axis ($m_e=1\mathrm{}$) because it occurs also in three dimensions, in contrast to the isotropic special one. The associated critical, crossover, and correction-to-scaling exponents are given to second order in $\epsilon$.