Semi-infinite anisotropic spherical model: Correlations atT>~Tc

Abstract

The ordinary surface magnetic phase transition is studied for the exactly solvable anisotropic spherical model ͑ASM͒, which is the limit D→ϱ 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, dр2, at finite temperatures. The correlation functions and the energy density profile in the semi-infinite ASM are calculated analytically and numerically for TуT c and 1рdрϱ. Since the lattice dimensionalities dϭ1, 2, 3, and 4 are special, a continuous spatial dimensionality dЈϭd Ϫ1 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 2ϽdϽ4 of Bray and Moore ͓Phys.

Ordering in magnetic films with surface anisotropy

“…An interesting feature of the solution for the surface-anisotropy model is that the Curie temperature of the bilayer becomes independent of the lattice structure in the Ising limit η s = 0. The result obtained above depends on the lattice dimensionality d only and, e.g., it is the same for the simple cubic model (d = 3) and the three-dimensional continuous-dimension model (d = 3.0) [6]. In the Ising limit, the lattice structure comes into play for trilayers and thicker films, where the inhomogeneity of the gap parameter G n becomes essential.…”

“…The inverse longitudinal correlation length κ z is determined by κ 2 z ≡ 2d[1/G−1] and it diverges at the critical point. In contrast to finite-D theories, where the longitudinal correlation length ξ cz ≡ 1/κ z plays the major role in the scaling, here in the limit D → ∞ it becomes only a slave variable, whereas all the physical quantities, except the longitudinal CF, are scaled with the transverse correlation length ξ c,⊥ ≡ 1/κ [6,4].…”

“…The lattice Green function P (X) satisfies P (0) = 1 and has a singularity at X → 1, the form of which in different dimensions can be found in Ref. [6]. For d ≤ 2, the Watson integral W ≡ P (1) goes to infinity; thus formula (2.21) yields nonzero values of the Curie temperature only for the anisotropic model, η < 1.…”

“…After the set of G n for a given temperature has been determined, one can compute the longitudinal CF σ zz nn ′ (q) from Eqs. In a usual situation, the above condition should be used in the middle of the film, n ∼ N/2, because for large N the critical divergence of the spin CF at the surface is suppressed [10,11,6]. If ordering of the film is driven by the surface, it is more convenient to use Eq.…”

“…In the spatially homogeneous bulk sample one has G n = G and η nn = η n,n±1 = η, and the transverse CF can be easily found [6,4]. The resulting equation for the gap parameter has the form θGP (ηG) = 1,…”