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 $\frac{1}{{r}^{d+\ensuremath{\sigma}}}$ (where $d$ is the dimensionality of the lattice and $\ensuremath{\sigma}>0$) is analyzed. It is shown that the model exhibits a ferromagnetic transition in one and two dimensions, providing $0<\ensuremath{\sigma} 2$. The asymptotic form of the spin-spin correlation function $\ensuremath{\Gamma}(\mathrm{r})$ is studied in the neighborhood of the critical temperature ${T}_{c}$. At $T={T}_{c}$, $\ensuremath{\Gamma}(\mathrm{r})$ decays for large $r$ as $\frac{1}{{r}^{d\ensuremath{-}\ensuremath{\sigma}}}$. Several two-dimensional models with long-range interactions falling off as $\frac{1}{{r}^{2}}$ in certain directions only are also investigated.