Renormalization Group and Critical Phenomena. II. Phase-Space Cell Analysis of Critical Behavior

A generalization of the Ising model is solved, qualitatively, for its critical behavior. In the generalization the spin S{n} at a lattice site n can take on any value from - to. The interaction contains a quartic term in order not to be pure Gaussian. The interaction is investigated by making a change of variable S{n}=m^₌ (n) S₌^', where the functions ₌ (n) are localized wavepacket functions. There are a set of orthogonal wave-packet functions for each order-of-magnitude range of the momentum k. An effective interaction is defined by integrating out the wave-packet variables with momentum of order 1, leaving unintegrated the variables with momentum 0. 5. Then the variables with momentum between 0. 25 and 0. 5 are integrated, etc. The integrals are computed qualitatively. The result is to give a recursion formula for a sequence of effective Landau-Ginsberg-type interactions. Solution of the recursion formula gives the following exponents: =0, =1. 22, =0. 61 for three dimensions. In five dimensions or higher one gets =0, =1, and =12, as in the Gaussian model (at least for a small quartic term). Small corrections neglected in the analysis may make changes (probably small) in the exponents for three dimensions.