New Phase Transitions of the Ising Model on Cayley Trees

Abstract

Abstract. We show that the nearest neighbors Ising model on the Cayley tree exhibits new temperature driven phase transitions. These transitions holds at various inverse temperatures different from the critical one. They are depicted by a change in the number of Gibbs states as well as by a drastic change of the behavior of free energies at these new transition points. We also consider the model in presence of an external field and compute the free energies of translation invariant periodic boundary conditions.Mathematics Subject Classifications (2010). 82B26 (primary); 60K35 (secondary)

Ising model on Cayley trees: a new class of Gibbs measures and their comparison with known ones

“…-measures corresponding to solutions on I 3 are weakly periodic, which coincide with the measures given in Theorem 1 for a 1 = k − |A|, a 2 = 0, a 3 = 0, a 4 = |A|, Fig. 5) Moreover, the system (3.8) was solved only in cases |A| = 1 and |A| = k (see [13], [14] [8]), Higuchi's non-translation-invariant measures (see [9]), Alternating Gibbs measures (see [6]) and weakly periodic measures for subgroups of index 4 (see Chapter 2 of [12] with more than 4 distinct values. Thus these measures are different from the measures mentioned in Theorem 1.…”

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

“…Note that if there is more than one positive fixed point of the operator (4.9), then there is more than one Gibbs measure corresponding to these positive fixed points. One says that a phase transition occurs for the Ising model, if the system of equations (4.5)-(4.8) has more than one solution [8,18,24]. The number of the solutions of equations (4.10) and (4.11) depends on the coupling constants and the parameter β = 1/T.…”

“…The theory of probability is one of the basic branches of mathematics lying at the base of the theory of statistical mechanics [8,[15][16][17][18][19][20]. As is known, one of the fundamental problems of statistical mechanics is to specify the set of all Gibbs measures associated to the given Hamiltonian [21][22][23][24][25]. A Gibbs measure is a probability measure frequently used in many problems of probability theory and statistical mechanics.…”

Free energies of the Potts model on a Cayley tree

“…k -periodic Gibbs measures, where G (2) k is the set of all words of even lengths. The corresponding set of vectors h = {h x ∈ R q−1 : x ∈ G k } has the form…”

“…The Cayley tree Γ k (See [2]) of order k ≥ 1 is an infinite tree, i.e., a graph without cycles, from each vertex of which exactly k + 1 edges issue. Let Γ k = (V, L, i) , where V is the set of vertices of Γ k , L is the set of edges of Γ k and i is the incidence function associating each edge l ∈ L with its endpoints x, y ∈ V .…”