Critical Region for the Ising Model with a Long-Range Interaction

Abstract

A version of the Ising model is developed in which the spin variables can be treated accurately in the continuum approximation. The perturbation series, both above and below the critical temperature $T_c$, is examined, and it is shown that there is a shift of $T_c$ from its mean-field value proportional to $q^{-2\mathrm{}}\ln q$, as well as the well-known shift proportional to $q^{-1\mathrm{}}$; here $q$ is the number of mutually interacting particles. It is shown, using renormalization theory, that there is a perturbation series in $q^{-1\mathrm{}}{|T-T_c|}^{-\frac{1}{2}}$ for which all terms are finite in the limit $q\to \infty$, if the shift of $T_c$ is put in correctly. For the two-dimensional model, the shift is shown to be proportional to $q^{-1\mathrm{}}\ln q$. Conditions are derived for a finite system to display critical behavior characteristic of three, two, one, or zero dimensions. It is shown how similar results can be obtained for a model similar to the Heisenberg model and for the standard Ising and Heisenberg models with interactions extending over many neighbors. A comparison is made between previously calculated numerical results for $T_c$ and the asymptotic forms derived here.