Phase diagram and extreme Gibbs measures of the Ising model on a Cayley tree in the presence of competing binary and ternary interactions

Ganikhodjaev, Nasir; Akın, Hasan; Uğuz, Selman; Temir, Seyit

doi:10.1080/01411594.2011.579395

Cited by 17 publications

(29 citation statements)

References 36 publications

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“…In previous works [4,5], for given Hamiltonians we obtained the Gibbs states that correspond in probability theory to what are called Markov chains with memory length 2. In this paper, we have studied the extreme Gibbs measures corresponding to Hamiltonian (1).…”

Section: Discussionmentioning

confidence: 99%

“…In this paper, we combine the results obtained in [4] and [5]. Our model with competing nearest-neighbors, next-nearest neighbors, and prolonged next-nearest-neighbors ternary interactions is dened by the following Hamiltonian:…”

Section: Introduction and Denitionsmentioning

confidence: 99%

“…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. In [5], we have obtained the extreme Gibbs measures of the Vannimenus model [6].…”

Section: Introduction and Denitionsmentioning

confidence: 99%

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

et al. 2013

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In this paper, we consider an Ising model with three competing interactions (nearest neighbor, next-nearest neighbor, and ternary prolonged neighbor) on the Cayley tree of order two, investigated by Ganikhodjaev et al. We study translation-invariant Gibbs measures of the Ising model with these competing interactions. Also, we investigate the set of the extreme Gibbs measures called Markov random elds with memory 2 of the model.

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Section: Discussionmentioning

confidence: 99%

Section: Introduction and Denitionsmentioning

confidence: 99%

Section: Introduction and Denitionsmentioning

confidence: 99%

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

et al. 2013

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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]. A Gibbs measure is a probability measure frequently used in many problems of probability theory and statistical mechanics.…”

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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“…1. For J 1 > 0 and k = 2 the phase diagram contains ferromagnetic phase and phase with period 3, that is new (see [8]). The interesting part of phase diagram corresponds to competing interactions (J 1 < 0).…”

Section: Phase Diagramsmentioning

confidence: 99%

Behaviors of Phase Diagrams of Ising Model with Competing Ternary and Binary Interactions on a Cayley Tree of Arbitrary Order

et al. 2012

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An Ising model with competing interactions has recently been studied extensively because of the appearance of nontrivial magnetic orderings. In this paper, we study the phase diagrams for the Ising model on a Cayley tree with competing nearest-neighbor interactions J and ternary prolonged interactions J tp on a Cayley tree of arbitrary order k and compare with the phase diagrams obtained in Uguz et al. and Vannimenus results for the Ising model on a Cayley tree with competing nearest-neighbor interactions J and ternary prolonged interactions J p . For some values of k, we obtain phase diagrams of the model. We clarify the role of order k of the Cayley tree. We also plot the variation of the wave vector with temperature.

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Phase diagram and extreme Gibbs measures of the Ising model on a Cayley tree in the presence of competing binary and ternary interactions

Cited by 17 publications

References 36 publications

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

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

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

Behaviors of Phase Diagrams of Ising Model with Competing Ternary and Binary Interactions on a Cayley Tree of Arbitrary Order

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