Break of universality for an Ising model with aperiodic Rudin-Shapiro interactions

Abstract

We analyze the ferromagnetic Ising model on non-Euclidean scale invariant lattices with aperiodic interactions (J A ,J B ,J C ,J D ) defined by Rudin-Shapiro substitution rules with Migdal-Kadanoff renormalization (MKR) and transfer matrix (TM) techniques. The analysis of the invariant sets of the zero-field MKR transformation indicates that the critical behavior, completely distinct from the one of the uniform model, is described by a new off-diagonal fixed point. This contrasts with other aperiodic models where the new critical behavior is described by a period-two cycle. With the new fixed point, values for the thermal critical exponents, $\alpha $ and $\nu$ , as well as the period of log-periodic oscillations, are obtained. Exact recursive maps for all thermodynamical functions are derived within the TM approach. The explicit dependence of the thermodynamical functions with respect to temperature is evaluated by the numerical iteration of the set of maps until a previously chosen convergence is achieved. They also indicate that, depending on the actual choice for the aperiodic coupling constants, the magnetic exponents ( $\beta $ and $\gamma$ ) assume different values. However the Rushbrook relation is always satisfied.