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

Abstract

A generalization of the Ising model is solved, qualitatively, for its critical behavior. In the generalization the spin ${S}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{n}}}$ at a lattice site $\stackrel{\ensuremath{\rightarrow}}{\mathrm{n}}$ can take on any value from $\ensuremath{-}\ensuremath{\infty} \mathrm{to} \ensuremath{\infty}$. The interaction contains a quartic term in order not to be pure Gaussian. The interaction is investigated by making a change of variable ${S}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{n}}}=\ensuremath{\Sigma}{m}^{}{\ensuremath{\psi}}_{m}(\mathrm{n}){S}_{m}^{\ensuremath{'}}$, where the functions ${\ensuremath{\psi}}_{m}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{n}})$ are localized wavepacket functions. There are a set of orthogonal wave-packet functions for each order-of-magnitude range of the momentum $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$. 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: $\ensuremath{\eta}=0$, $\ensuremath{\gamma}=1.22$, $\ensuremath{\nu}=0.61$ for three dimensions. In five dimensions or higher one gets $\ensuremath{\eta}=0$, $\ensuremath{\gamma}=1$, and $\ensuremath{\nu}=\frac{1}{2}$, 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.