On Free Energies of the Ising Model on the Cayley Tree

Gandolfo, Daniel; Рахматуллаев, Музаффар Мухаммаджанович; Rozikov, U. A.; Ruiz, Jean

doi:10.1007/s10955-013-0713-0

Cited by 34 publications

(23 citation statements)

References 11 publications

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“…In [34,43], the authors have presented, for the Ising model on the Cayley tree, some explicit formulae of the free energies (and entropies) according to boundary conditions (b.c.). By applying the general formulae to various known boundary conditions on arbitrary order chandelier-lattices, we plan to obtain some explicit formula of free energy and relative entropy corresponding to the boundary conditions in our future work.…”

Section: Discussionmentioning

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: Discussionmentioning

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 [1] for the Ising model (with the set {−1, 1} of spin values) the authors constructed a class of new Gibbs measures by extending the known Gibbs measures defined on a Cayley tree of order k 0 to a Cayley tree of higher order k > k 0 . Their construction is called ART-construction [7].…”

Section: Non-translation-invariant Gibbs Measuresmentioning

confidence: 99%

Nontranslation invariant Gibbs measures for models with uncountable set of spin values on a Cayley tree

Rozikov¹,

Botirov²

2018

Reports on Mathematical Physics

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We consider models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ≥ 1. It is known that the "splitting Gibbs measures" of the model can be described by solutions of a nonlinear integral equation. Recently, solving this integral equation some periodic (in particular translation-invariant) splitting Gibbs measures were found. In this paper we give three constructions of new sets of non-translation-invariant splitting Gibbs measures. Our constructions are based on known solutions of the integral equation.Mathematics Subject Classifications (2010). 82B05, 82B20 (primary); 60K35 (secondary)

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“…1 Adding a boundary field at each site of the boundary is called a generalized boundary condition [3] or boundary law [6] Now by (2.2) and (3.2)-(3.4) we get…”

Section: Translation-invariant Limiting Gibbs Measuresmentioning

confidence: 99%

Boundary Conditions for Translation-Invariant Gibbs Measures of the Potts Model on Cayley Trees

Gandolfo

Рахматуллаев²,

Rozikov³

2017

J Stat Phys

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Abstract. We consider translation-invariant splitting Gibbs measures (TISGMs) for the qstate Potts model on a Cayley tree of order two. Recently a full description of the TISGMs was obtained, and it was shown in particular that at sufficiently low temperatures their number is 2 q − 1. In this paper for each TISGM µ we explicitly give the set of boundary conditions such that limiting Gibbs measures with respect to these boundary conditions coincide with µ.Mathematics Subject Classifications (2010). 82B26; 60K35.

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On Free Energies of the Ising Model on the Cayley Tree

Cited by 34 publications

References 11 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

Nontranslation invariant Gibbs measures for models with uncountable set of spin values on a Cayley tree

Boundary Conditions for Translation-Invariant Gibbs Measures of the Potts Model on Cayley Trees

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