A study of the asymptotic behavior of the D-dimensional Ising model

Abstract

A general method is introduced for studying the asymptotic behavior of the D -dimensional Ising model. In our approach, the continuum representation of the Ising partition function is analyzed by using bifurcation theory in conjunction with well-known asymptotic methods of analysis. Following a suggestion of Gallavotti, it is shown rigorously that in the range H ( = J/kT) < 1 4 D the gaussian model yields the correct asymptotic behavior of the Ising model, and there is no phase transition in this range. For the range H > 1 4 D, the behavior of the Ising model is obtained bu “bifurcating” from the gaussian model, and the solution in this range is found to be very similar in structure to the one obtained for the corresponding D-dimensional spherical model. We find bifurcation in all dimensions, no phase transition in one-dimension, but a phase transition in all D ⩾ 2. For D = 2, the phase transition is characterized by a logarithmic singularity in the specific heat (in accordance with the exact behavior found by Onsager). For D = 3, the transition is characterized by an algebraic singularity in the specific heat (in particular, the singularity is (λ - λ c) -1 2 where λ is a parameter related to, but not identical with, the temperature). With regard to quantitative estimates, since in this paper the bifurcation analysis was carried through only to order ε 2, where ε is the bifurcation parameter, our estimates of the transition temperature in 2 and 3 dimensions are somewhat displaced from the accepted values. Finally, the possible relevance of our approach to other problems of interest in the theory of phase transitions is indicated.