Critical Behavior of Ferromagnetic Spin Models with Aperiodic Exchange Interactions

Abstract

We review recent investigations of the critical behavior of ferromagnetic q-state Potts models on a class of hierarchical lattices, with exchange interactions according to some deterministic but aperiodic substitution rules. The problem is formulated in terms of exact recursion relations on a suitable parameter space. The analysis of the xed points of these equations leads to a criterion to gauge the relevance of the aperiodic geometric uctuations. For irrelevant uctuations, the critical behavior remains unchanged with respect to the underlying uniform models. In the presence of relevant uctuations, a non-trivial symmetric xed point, associated with the critical behavior of the uniform model, becomes fully unstable, and there appears a two-cycle of the recursion relations. A scaling analysis, supported by direct numerical thermodynamical calculations, shows the existence of a novel critical universality class associated with relevant geometric uctuations.