Migdagkadanoff Study of Z, Symmetric Systems with Generalized Action*

The real space renormalization group is a powerful and elegant tool for the study of phase transitions in statistical systems. Concepts such as the universality of critical exponents are readily explained in terms of renormalization group flows in the vicinity of fixed points. In this paper we discuss how generalized Z, models provide a simple illustration of these ideas in a two-parameter coupling constant space. For an exact treatment, one should consider renormalization group equations for an infinite number of coupling constants, involving both nearest and non-nearest neighbor interactions. For practical calculation some truncation to a few couplings is necessary. The Migdal-Kadanoff recursion relations provide such a truncation [l]. Although the errors involved are difficult to assess, the approximation is particularly simple and makes specific predictions for critical temperatures and exponents. For gauge theories the relations predict the critical nature of four dimensions. Indeed, before the advent of Monte Carlo calculations [2], this was the strongest evidence for quark confinement in the non-Abelian gauge theory of the strong interactions. Unfortunately, the recursion relations often make incorrect predictions on the order of the gauge theory transitions. Nevertheless, they appear to correctly predict when a phase transition is to be expected. This, plus the simplicity of the method, makes it worthy of further study.