MODULATED STRUCTURES IN THE ISING-MODEL WITH COMPETING INTERACTIONS ON THE CAYLEY TREE

We introduce two analogs of the axial next-nearest-neighbor Ising (ANNNI) model on a Cayley tree. Besides the competing interactions along the branches of the tree, we include a set of pair interactions on the same generation to mimic the ferromagnetic layers of the ANNNI model. In the infinite-coordination limit, the statistical problem is formulated as a discrete, nonlinear, two-dimensional map. The phase diagrams display a Lifshitz multicritical point and many sequences of modulated structures characterized by a principal wave number. At low temperatures, we perform numerical calculations to show the existence of complete devil's staircases. At higher temperatures, the incommensurate structures occupy finite portions of the phase diagrams. We discuss the existence of pinned incommensurate phases. Also, we calculate the Lyapunov exponents of the map to show the presence of chaotic structures, associated with a strange attractor. The main modulated phase of the long-range model under consideration displays a transition between characteristic structures at low and high temperatures.