Spherical Model with Long-Range Ferromagnetic Interactions

Abstract

The behavior of the spherical model of a ferromagnet with an interaction energy between the magnetic spins which varies with distance as $\frac{1}{r^{d+\sigma }}$ (where $d$ is the dimensionality of the lattice and $\sigma >0\mathrm{}$) is analyzed. It is shown that the model exhibits a ferromagnetic transition in one and two dimensions, providing $0<\mathrm{}\sigma <d$. (The usual spherical model with nearest-neighbor interactions does not have a transition in one and two dimensions.) The critical-point behavior is investigated. It is found that the singularities in the specific heat and susceptibility are dependent on $\sigma$ and $d$, but the behavior of the magnetization is independent of $\sigma$ and $d$. In three dimensions the susceptibility diverges as ${(T-T_c)}^{-\gamma }$, where $\gamma =1\mathrm{}$ for $0<\mathrm{}\sigma <\frac{3}{2}$, $\gamma =\frac{\sigma }{(3-\sigma )}$ for $\frac{3}{2}<\sigma <2\mathrm{}$ and $\gamma =2\mathrm{}$ for $\sigma >2\mathrm{}$. The asymptotic form of the spin-spin correlation function $\Gamma \left(\mathrm{r}\right)$ is studied in the neighborhood of the critical temperature $T_c$. At $T=T_c$, $\Gamma \left(\mathrm{r}\right)$ decays for large $r$ as $\frac{1}{r^{d-\sigma }}$. Several two-dimensional models with long-range interactions falling off as $\frac{1}{r^2}$ in certain directions only are also investigated.