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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 T₂, is examined, and it is shown that there is a shift of T₂ 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-{T₂|}^-1{2} for which all terms are finite in the limit q, if the shift of T₂ 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 T₂ and the asymptotic forms derived here.
D. J. Thouless (Sat,) studied this question.