Semi-infinite anisotropic spherical model: Correlations at $T>~T_c$

The ordinary surface magnetic phase transition is studied for the exactly solvable anisotropic spherical model (ASM), which is the limit $\overrightarrow{D}\infty$ of the $D$-component uniaxially anisotropic classical vector model. The bulk limit of the ASM is similar to that of the spherical model, apart from the role of the anisotropy stabilizing ordering for low lattice dimensions, $d<~2$, at finite temperatures. The correlation functions and the energy density profile in the semi-infinite ASM are calculated analytically and numerically for $T>~T_c$ and $1<~d<~\infty$. Since the lattice dimensionalities $d=1\mathrm{}$, 2, 3, and 4 are special, a continuous spatial dimensionality $d^{\prime }=d-1$ has been introduced for dimensions parallel to the surface. However, preserving a discrete layer structure perpendicular to the surface avoids unphysical surface singularities and allows numerical solitions that reveal significant short-range features near the surface. The results obtained generalize the isotropic-criticality results for $2<d<\mathrm{}4\mathrm{}$ of Bray and Moore [Phys. Rev. Lett. 38, 735 (1977); J. Phys. A 10, 1927 (1977)].