The Hamiltonian three-parameter spaces of the q-state Potts model with three competing interactions on a Cayley tree

Abstract

In this paper, we study the phase diagrams for the integer q-state (q ≥ 3) Potts model on a Cayley tree for order two with competing nearest-neighbor interactions J 1, prolonged next-nearestneighbor interactions J p and two-level triple-neighbor interactions J t . The exact phase diagrams of the Potts model with some competing interactions on a Cayley tree lattice of order two have been found. At vanishing temperature, the phase diagram is fully determined for all values and signs of J 1, J p and J t . Our aim is to generalize the results of Ganikhodjaev et al. to the q-state Potts model with competing nearest-neighbor, prolonged next-nearest-neighbor and two-level tripleneighbor interactions on a Cayley tree for order 2 and to compare these with previous results in the literature. Ganikhodjaev et al. reported on a new phase, denoted as a paramodulated (PM) phase, found at low temperatures and characterized by 2-periodic points of an one-dimensional dynamical system lying inside the modulated phase. An important note for such a phase is that inherently the Potts model has no analogues in the Ising setting. In this paper, we show that increasing the spin number from three (q = 3) to arbitrary q > 3 can dramatically affect the resultant phases (expanding the paramodulated phase). We believe that the enlarging of the paramodulated (PM) phase is essentially connected to the symmetry in the spin numbers.