Phase diagrams of two-dimensionalZ(q)models

The phase diagrams of general $q$-state planar spin models on a square lattice are studied for $q=4-7\mathrm{}$. The phase diagrams are calculated by a variational method and the infinitesimal Migdal renormalization-group (RG) transformations. Since the Migdal RG commutes with the self-dual transformations of $Z\mathrm{}\left(q\right)\mathrm{}$ models, one expects reliable phase diagrams (but not accurate critical exponents). In fact all exactly known critical points are reproduced by our calculations together with a fair numerical simulation of lines of critical points. Our results clearly indicate the existence of massless phases for $q=6\mathrm{} \mathrm{and} 7$ but are inconclusive for $q=5\mathrm{}$. More accurate methods are needed to determine the phase structure of the $Z\mathrm{}\left(5\right)\mathrm{}$ model; see, for example, the finite-lattice approximation of Roomany and Wyld (preceding paper).