On Pure Phases of the Vannimenus Model

On phase transitions of the Potts model with three competing interactions on Cayley tree

“…J, J 1 , J 2 ∈ R are coupling constants and h is external field. In [15], the authors have already obtained the exact solutions of the Potts model with competing ternary and binary interactions and external field on Cayley tree described by means of the Hamiltonian (2.2), where h = 0 and J 1 = 0. In this paper, we assume that each parameter J, J 1 , J 2 , h is different from zero.…”

“…Instead of the configurations σ(V n ) and the partition functions Z Vn in volume V n we use notations σ n and Z (n) , respectively [10,15]. It is well known that the Gibbs measure is a probability measure used in many problems of probability theory and of statistical mechanics [13].…”

“…. , q}, q > 2 which are associated with each vertex of the tree J k (see [15] for details). The Potts model with three competing interactions and external field is defined by the following Hamiltonian…”

“…We generalize the problem in [2] for the Potts model with three competing interactions and non-zero external field. This problem has been investigated for the Potts model with only two competing interactions [2,3].…”

Description of Extreme Gibbs Measures for the Ising Model with Three Interactions

“…It is well known that such measures form a nonempty convex compact subset in the set of all probabilistic measures. The purpose of this paper is to investigate the Gibbs measures of the Ising model [3] with ternary prolonged and nearest neighbor interactions on the Cayley tree of order two and to describe its extreme elements (pure phases). In [4], we have studied phase diagram and extreme Gibbs measures of the Ising model on a Cayley tree in the presence of competing binary and ternary interactions.…”