Phase diagram of Ising model on Cayley tree with competing interactions up to third nearest-neighbor generation

Abstract

We study the phase diagram of the Ising model on Cayley tree with competing the first-, second-, and third nearest-neighbor interactions. Ising model was described as an easily defined class of statistical mechanic of lattice models. This study addresses the phase diagram of Ising model on a Cayley tree, with competing interactions up to the third nearest-neighbor generations. For Ising model on Cayley tree derived a recurrence system of equations in the same line of Vannimenus’s approach, and its described phase diagram. The phase diagrams were obtained from stability conditions, and characteristic points in the iteration scheme were numerically analyzed. The main novelty of this thesis is it considers an Ising model on the Cayley tree with competing interactions up to third nearest-neighbor generation with spins belonging to different branches of the tree. In addition to the expected ferromagnetic, anti-ferromagnetic and paramagnetic phases, we present a new paramodulated phase. Furthermore, we also study the phase diagrams for the Ising model that defined on Cayley tree-like lattice: pentagonal chandelier, with competing one-level pentagon interactions. Pentagonal chandelier can be viewed as another geometrical representation of Cayley tree order 5. Finally, we study the variation of wavevector with temperature in the modulated phase in detail where narrow commensurate steps between incommensurate regions appeared when investigating the Lyapunov exponent associated with trajectory of the system.