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

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 Tc, is examined, and it is shown that there is a shift of Tc from its mean-field value proportional to q^-2lnq, as well as the well-known shift proportional to q^-1; here q is the number of mutually interacting particles. It is shown, using renormalization theory, that there is a perturbation series in q^-1|T-Tc|^-1/2 for which all terms are finite in the limit q→∞, if the shift of Tc is put in correctly. For the two-dimensional model, the shift is shown to be proportional to q^-1lnq. 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 Tc and the asymptotic forms derived here.