Lyapunov Exponents and Modulated Phases of an Ising Model on Cayley Tree of Arbitrary Order

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

doi:10.1142/s0129183112500398

Cited by 10 publications

(41 citation statements)

References 24 publications

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“…The recurrence equations obtained in the present paper totally differ from [1,28,34,39]. Note that for the Ising model associated with the Hamiltonian (3.4) on the chandelier lattices of order k, in contrast to the symmetry of arbitrary order Cayley tree [6,39], if k > 3, then the chandelier lattice of order k is not symmetry. Therefore, in order to construct the recurrence equations associated with the given Hamiltonian (3.4) for k > 3 is much more difficult.…”

Section: Discussioncontrasting

confidence: 53%

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: Discussioncontrasting

confidence: 53%

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

Akιn¹

2019

Condens. Matter Phys.

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“…In [22], we obtain a new set of limiting Gibbs measures for the Ising model on a Cayley tree. In [2,17,18], the authors study the phase diagram for the Ising model on a Cayley tree of arbitrary order k with competing interactions. In [18], the authors characterized each phase by a particular attractor and the obtained the phase diagram by following the evolution and detecting the qualitative changements of these attractors.…”

Section: Introductionmentioning

confidence: 99%

Gibbs measures with memory of length 2 on an arbitrary-order Cayley tree

Akın¹

2018

Int. J. Mod. Phys. C

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In this paper, we consider the Ising-Vanniminus model on an arbitrary order Cayley tree. We generalize the results conjectured in [3,4] for an arbitrary order Cayley tree. We establish existence and a full classification of translation invariant Gibbs measures with memory of length 2 associated with the model on arbitrary order Cayley tree. We construct the recurrence equations corresponding generalized ANNNI model. We satisfy the Kolmogorov consistency condition. We propose a rigorous measure-theoretical approach to investigate the Gibbs measures with memory of length 2 for the model. We explain whether the number of branches of tree does not change the number of Gibbs measures.Also we take up with trying to determine when phase transition does occur.

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“…In addition, models such as Ising and Potts on the Cayley tree can be helpful in discovering additional systems with related properties. As a result, many researchers have employed the Ising and Potts models in conjunction with the Cayley tree (Bethe lattices) [4,5,6,7,9,17,18,33]. The Ising model has relevance to physical, chemical, and biological systems [12,21,22,23].…”

Section: Introductionmentioning

confidence: 99%

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

Akın

2016

Chinese Journal of Physics

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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.

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Lyapunov Exponents and Modulated Phases of an Ising Model on Cayley Tree of Arbitrary Order

Cited by 10 publications

References 24 publications

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

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

Gibbs measures with memory of length 2 on an arbitrary-order Cayley tree

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

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