Ising model on a Cayley tree with competing and aperiodic interactions

Abstract

We include extra aperiodic interactions in the Ising model with competing ferromagnetic and antiferromagnetic interactions between first and second neighbors along the branches of a Cayley tree. The problem is formulated as a nonlinear map whose attractors correspond to solutions deep in the interior of the tree. We present some analytical and numerical calculations for aperiodic interactions introduced according to the rules of a Koch curve and a Fibonacci sequence. For small values of a parameter of aperiodicity, there are no relevant changes in the phase diagram, except for the appearance of some bumps along the paramagneticmodulated transition. As the aperiodic interactions become more important, there are several new phenomena, such as phase locking and phase splitting, and the enhancement of the regions of chaotic attractors and of co-stability of different structures. @S1063-651X~97!14307-0#