Renormalization Group and Critical Phenomena. II. PhaseSpace Cell Analysis of Critical Behavior
Abstract
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>}=mψ_{m}(n)S'_{m}, where the functions ψ_{m}(n>) are localized wavepacket functions. There are a set of orthogonal wavepacket functions for each orderofmagnitude range of the momentum k>. An effective interaction is defined by integrating out the wavepacket 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 LandauGinsbergtype interactions. Solution of the recursion formula gives the following exponents: η=0, γ=1.22, ν=0.61 for three dimensions. In five dimensions or higher one gets η=0, γ=1, and ν=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.
 Publication:

Physical Review B
 Pub Date:
 November 1971
 DOI:
 10.1103/PhysRevB.4.3184
 Bibcode:
 1971PhRvB...4.3184W