Phase diagrams of the semi-infiniteZ(q) models by real space renormalization-group method

Abstract

We investigate the three-dimensional semi-infiniteZ(q) models using an infinitesimal Migdal-Kadanoff method. A rich variety of phase diagrams is obtained. The massless spin-wave phase which appears in the infinite two-dimensionalZ(q) models on the Clock line between the disordered and ferromagnetic phase forq≧q _c is also present in the semi-infinite system on the surface when the bulk is disordered. We also observe that if the bulk is in the phase to which the symmetry is engendered by a subgroupZ(p), such thatp<q, the surface of the system is in the same phase or in a less symmetrical phase to which the symmetry is engendered by a subgroupZ(m) ofZ(p) such thatp=αm (m≦p) with α an integer number satisfying 1≦α≦p. The case α=p corresponds to the least symmetrical phase. Since the infiniteZ(q) models exhibit a rich variety of phase transitions and multicritical points, the semi-infinite models present new ordinary, extraordinary, surface and special phase transitions which do not occur in the semi-infinite Ising-like systems. As theZ(q) model transforms into theX−Y model whenq→∞, we have deduced the phase diagram of the semi-infiniteX−Y model. It is qualitatively similar to the phase diagram of the semi-infinite Ising model.