Using new approaches to obtain Gibbs measures of Vannimenus model on a Cayley tree

Abstract

Abstract. In this paper, we consider Vannimenus model with competing nearest-neighbors and prolonged next-nearest-neighbors interactions on a Cayley tree. For this model we define Markov random fields with memory of length 2. By using a new approach, we obtain new sets of Gibbs measures of Ising-Vannimenus model on Cayley tree of order 2. We construct the recurrence equations corresponding Ising-Vannimenus model. We prove the Kolmogorov consistency condition. We investigate the translation-invariant and periodic non transition-invariant Gibbs measures for the model. We find new sets of Gibbs measures different from the Gibbs measures given in the references [20,25]. We show that some of the measures are extreme Gibbs distributions.

Gibbs measures of an Ising model with competing interactions on the triangular chandelier-lattice

“…It is well known that the comprehension of phase transitions is one of the most interesting, perhaps the central, problems of equilibrium statistical mechanics [7]. By the phase transition we mean the existence of at least two distinct Gibbs measures associated with the given model [1,8,18,28]. We will investigate the existence of translation invariant Gibbs measures on a wide class of the TCL, restricted only to the memory of length 2.…”

“…Previously, researchers frequently used memory of length 1 over a Cayley tree to study Gibbs measures [31]. In [1,21,22,34], the authors have studied Gibbs measures with a memory of length 2 for generalized ANNNI models on a Cayley tree of order 2 by means of a vector valued function (see [35] for details). In [1,21], the next generalizations are considered.…”

“…In [1,21,22,34], the authors have studied Gibbs measures with a memory of length 2 for generalized ANNNI models on a Cayley tree of order 2 by means of a vector valued function (see [35] for details). In [1,21], the next generalizations are considered. These authors have defined Gibbs measures or Gibbs states with a memory of length 2 (on spin-configurations σ) for generalized ANNNI models on Cayley trees of order 2.…”

“…× exp abc j h B 1 (y (1) ), j abc + de f k h B 1 (y (2) ),kde f + glmn h B 1 (y (3) ),lgmn . (3.9) Here, i, j, k, l ∈ {−1, +1} and y (1) , y (2) , y (3) ∈ S(x (0) ) (see figure 3). Therefore, we have obtained an explicit formula for Gibbs measures with memory of length 2 satisfying consistency conditions by means of equation (3.9).…”

“…As is known, Cayley tree (or Bethe lattice), introduced by Hans Bethe in 1935, is a non-realistic lattice. Since other operations and the calculations on this lattice are easier to understand than the ddimensional Z d lattice, many of the topics in statistical physics have recently been taken into account on the Cayley tree [1]. Thus, the results obtained on the Cayley tree became a source of inspiration for the d-dimensional Z d lattice.…”