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 $S_{\overrightarrow{\mathrm{n}}}$ at a lattice site $\overrightarrow{\mathrm{n}}$ can take on any value from $-\infty \mathrm{to} \infty$. The interaction contains a quartic term in order not to be pure Gaussian. The interaction is investigated by making a change of variable $S_{\overrightarrow{\mathrm{n}}}=\Sigma m^{}{\psi }_m\left(\mathrm{n}\right)S_m^{\prime }$, where the functions ${\psi }_m\left(\overrightarrow{\mathrm{n}}\right)$ are localized wavepacket functions. There are a set of orthogonal wave-packet functions for each order-of-magnitude range of the momentum $\overrightarrow{\mathrm{k}}$. 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: $\eta =0\mathrm{}$, $\gamma =1.22\mathrm{}$, $\nu =0.61\mathrm{}$ for three dimensions. In five dimensions or higher one gets $\eta =0\mathrm{}$, $\gamma =1\mathrm{}$, and $\nu =\frac{1}{2}$, 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.