Semi-Infinite Anisotropic Spherical Model: Correlations at T >= T_c

TL;DR

This paper analytically and numerically investigates surface correlations in the anisotropic spherical model at and above the critical temperature, extending known isotropic criticality results to anisotropic cases across various dimensions.

Contribution

It provides a detailed analysis of surface correlations in the anisotropic spherical model, including numerical solutions that account for anisotropy and dimensionality effects, generalizing previous isotropic results.

Findings

Correlation functions are analytically derived for T >= T_c.

Numerical solutions reveal significant short-range surface features.

Results extend isotropic criticality findings to anisotropic models across dimensions.

Abstract

The ordinary surface magnetic phase transition is studied for the exactly solvable anisotropic spherical model (ASM), which is the limit D \to \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 dimensionalities, 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=1,2,3, and 4 are special, a continuous spatial dimensionality d'=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 < 4 of Bray and Moore [Phys. Rev. Lett. 38, 735 (1977); J. Phys. A: Math. Gen. 10, 1927 (1977)].