A New Set of Limiting Gibbs Measures for the Ising Model on a Cayley Tree

Akın, Hasan; Rozikov, U. A.; Temir, Seyit

doi:10.1007/s10955-010-0106-6

Cited by 40 publications

(35 citation statements)

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“…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].…”

Section: Introductionmentioning

confidence: 99%

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

Akιn¹

2019

Condens. Matter Phys.

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In this paper, we consider an Ising model with three competing interactions on a triangular chandelier-lattice (TCL). We describe the existence, uniqueness, and non-uniqueness of translation-invariant Gibbs measures associated with the Ising model. We obtain an explicit formula for Gibbs measures with a memory of length 2 satisfying consistency conditions. It is rigorously proved that the model exhibits phase transitions only for given values of the coupling constants. As a consequence of our approach, the dichotomy between alternative solutions of Hamiltonian models on TCLs is solved. Finally, two numerical examples are given to illustrate the usefulness and effectiveness of the proposed theoretical results. Figure 1. (Colour online) Cayley tree-like lattice: triangular chandelier with 2 level. Three successive generations of TCL (J represents nearest-neighbour interactions; J p represents prolonged next nearestneighbour interactions and J SL represents same-level nearest-neighbours interactions.

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Section: Introductionmentioning

confidence: 99%

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

Akιn¹

2019

Condens. Matter Phys.

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“…It is also well known that the structure of the lattice plays an important role in examining the spin systems. The Potts model, as a generalization of the Ising model on Cayley tree with competing interactions appeared in a pioneering work of Vannimenus [4], has recently been studied extensively (see [1,3,5,6]). On the Cayley tree one can consider two types of next-nearest-neighbors (triple neighbors): prolonged and one-level next-nearest-neighbors (respectively two-level triple neighbors).…”

Section: Introductionmentioning

confidence: 99%

“…The approach used to study Gibbs measure on Cayley tree is based on the Markov random field on trees and recurrent equations of this theory [2][3][4][5]. In [4], we have analytically studied the recurrence equations of a generalized (Axial Next-Nearest-Neighbor Ising) ANNNI model on a Cayley tree and obtained some exact results: critical temperatures and curves, number of phases, partition function.…”

Section: Introductionmentioning

confidence: 99%

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On Gibbs Measures of the Potts Model with Three Competing Interactions on Cayley Tree of Order 3

Akın

Saygili

2016

Acta Phys. Pol. A

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In this paper, we consider the Potts model with competing interactions on the Cayley tree of order three. We give the Potts model on the Cayley tree and its recursion relation. We construct the Gibbs states corresponding to the model by using Markov random field method. We calculate the critical curve, such that there is a phase transition for the model. We show that there are phase transition of the model for some given parameters. We extend the results obtained by Akin and Temir (Condensed Matt.

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“…In the previous section we have shown that the strong phase transition occurs on a Cayley tree of order three. In this section we are going to describe new p-adic generalized Gibbs measures of the Ising model on the Cayley tree of order k ≥ 3 by method ART (see [27]). We recall that each solution of (3.5) define a p-adic generalized Gibbs measures for Ising model on the Cayley tree of order k ≥ 1.…”

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confidence: 99%