Spherical model and the infinite spin dimensionality limit

The ordinary surface magnetic phase transition is studied for the exactly solvable anisotropic spherical model (ASM), which is the limit $\stackrel{\ensuremath{\rightarrow}}{D}\ensuremath{\infty}$ 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 $1<~d<~\ensuremath{\infty}$. Since the lattice dimensionalities $d=1$ , 2, 3, and 4 are special, a continuous spatial dimensionality ${d}^{\ensuremath{'}}=d\ensuremath{-}1$ 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 10, 1927 (1977)].The ordinary surface magnetic phase transition is studied for the exactly solvable anisotropic spherical model (ASM), which is the limit $\stackrel{\ensuremath{\rightarrow}}{D}\ensuremath{\infty}$ 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 $1<~d<~\ensuremath{\infty}$. Since the lattice dimensionalities $d=1$, 2, 3, and 4 are special, a continuous spatial dimensionality ${d}^{\ensuremath{'}}=d\ensuremath{-}1$ 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 10, 1927 (1977)].

arXiv:cond-mat/9803230Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsJul 1, 19986 citations

The Spherical Model of a Ferromagnet

T. H. Berlin, Mark Kac

A mathematical model, the spherical model, of a ferromagnet is described. The model is a generalization of the Ising model; and one-, two-, and three-dimensional lattices of infinite extent can be extensively discussed. A three-dimensional lattice shows ferromagnetic behavior and provides a statistical model of the Weiss phenomenological theory. The limiting free energy appears in a form which contains two of the essential features of the exactly known Ising model results in one and two dimensions. This suggests the probable form of the limiting free energy for the three-dimensional Ising model. A simplified model, the Gaussian model, is briefly discussed because this model also contains some of the significant features of the Ising model. However, the Gaussian model, unlike the spherical model, is not defined for all temperatures.A mathematical model, the spherical model, of a ferromagnet is described. The model is a generalization of the Ising model; and one-, two-, and three-dimensional lattices of infinite extent can be extensively discussed. A three-dimensional lattice shows ferromagnetic behavior and provides a statistical model of the Weiss phenomenological theory. The limiting free energy appears in a form which contains two of the essential features of the exactly known Ising model results in one and two dimensions. This suggests the probable form of the limiting free energy for the three-dimensional Ising model. A simplified model, the Gaussian model, is briefly discussed because this model also contains some of the significant features of the Ising model. However, the Gaussian model, unlike the spherical model, is not defined for all temperatures.

Physical ReviewJun 15, 19521,028 citations

Distribution of the total spin in the spherical model with long-range potential

Leonid Victorovich Bogachev

Theoretical and Mathematical PhysicsMar 1, 19782 citations

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 $1<~d<~\ensuremath{\infty}$. Since the lattice dimensionalities

d=1

, 2, 3, and 4 are special, a continuous spatial dimensionality ${d}^{\ensuremath{'}}=d\ensuremath{-}1$ 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 10, 1927 (1977)].