A Class of One-Dimensional Models with Unique Ground States that Admits Phase Transitions

Abstract

We will construct a class of one-dimensional models with unique ground states and two spin variables that have a countable number of extreme limits Gibbs states. The work is a. modification and extension of some previous work. We will show that it is an improvement over such work.

Gibbs measures of an Ising-Vannimenus Model with one-level competing interactions on 4th order Cayley tree

“…Gibbs measure and infinite random systems were introduced in 's by Dobrushin who introduced the Markov random fields on the d-dimensional lattices [ 1 ]. During the last few decades many authors dealt with the problem of Gibbs measures and phase transitions [ [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21] , [22] , [23] , [24] , [25] , [26] , [27] , [28] , [29] , [30] , [31] , [32] , [33] , [34] , [35] , [36] , [37] , [38] , [39] , [40] ].…”

“…Bethe lattice (or Cayley tree) is not a realistic lattice, it was introduced in the physical literature in 1935 by the physicist Hans Bethe [ 18 ]. Many authors have considered the well-known models, the Ising and Potts models, on the Cayley tree [ [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [23] , [24] , [25] , 30 , [34] , [35] , [36] , [37] , [38] , [39] , [40] ].…”