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

Akιn,

doi:10.5488/cmp.22.23002

Cited by 11 publications

(17 citation statements)

References 42 publications

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“…We plan to experiment in our future work in order to analyse the same problem analytically for the high-order Cayley 13001-13 tree. Recently, the author [47] has studied the existence of the Gibbs measures of an Ising model with competing interactions on the triangular chandelier-lattice. The phase diagrams corresponding to the Gibbs states on the Cayley-like lattices have not been examined yet.…”

Section: Discussionmentioning

confidence: 99%

Determination of paramagnetic and ferromagnetic phases of an Ising model on a third-order Cayley tree

Akın¹

2021

Condens. Matter Phys.

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In this present paper, the recurrence equations of an Ising model with three coupling constants on a third-order Cayley tree are obtained. Paramagnetic and ferromagnetic phases associated with the Ising model are characterized. Types of phases and partition functions corresponding to the model are rigorously studied. Exact solutions of the mentioned model are compared with the numerical results given in Ganikhodjaev et al. [J. Concr. Appl. Math., 2011, 9, No. 1, 26-34].

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Section: Discussionmentioning

confidence: 99%

Determination of paramagnetic and ferromagnetic phases of an Ising model on a third-order Cayley tree

Akın¹

2021

Condens. Matter Phys.

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“…We will examine similar problems such as entropy and ergodic properties of these LCAs on the chandelier network. In [103], we investigated the Gibbs measures of an Ising model with competing interactions on the TCL. Gibbs measures on the other chandelier networks have not yet been explored.…”

Section: -37mentioning

confidence: 99%

Phase diagrams of lattice models on Cayley tree and chandelier network: a review

Akın¹

2022

Condens. Matter Phys.

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The main purpose of this review paper is to give systematically all the known results on phase diagrams corresponding to lattice models (Ising and Potts) on Cayley tree (or Bethe lattice) and chandelier networks. A detailed survey of various modelling applications of lattice models is reported. By using Vannimenus's approach, the recursive equations of Ising and Potts models associated to a given Hamiltonian on the Cayley tree are presented and analyzed. The corresponding phase diagrams with programming codes in different programming languages are plotted. To detect the phase transitions in the modulated phase, we investigate in detail the actual variation of the wave-vector q with temperature and the Lyapunov exponent associated with the trajectory of our current recursive system. We determine the transition between commensurate (C) and incommensurate (I) phases by means of the Lyapunov exponents, wave-vector, and strange attractor for a comprehensive comparison. We survey the dynamical behavior of the Ising model on the chandelier network. We examine the phase diagrams of the Ising model corresponding to a given Hamiltonian on a new type of "Cayley-tree-like lattice", such as triangular, rectangular, pentagonal chandelier networks (lattices). Moreover, several open problems are discussed.

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“…Proof. From the simple property of the logarithm and exponentiation functions, it is easy to show the existence of the limit given in Equation (17).…”

Section: Lemmamentioning

confidence: 99%

“…This section deals with the existence of the phase transition of the model by means of the self-similarity method. Recently, many papers have discussed the phase transition problem of the given lattice models using the KCM method [4,[16][17][18].…”

Section: Limiting Gibbs Measures and The Phase Transitionmentioning

confidence: 99%

“…To the best of the author's knowledge, the free energy formulas of the lattice models on the CT were calculated using the KCM [4,10,14,15]. Recently, the Gibbs measures of lattice models on CT-like lattices and the consequent phase transition problem have been investigated using the KCM (see [16,17]). The Gibbs measures for the q-state Potts model on the CT were determined using the self-similarity method [11,18].…”

Section: Introductionmentioning

confidence: 99%

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Calculation of the Free Energy of the Ising Model on a Cayley Tree via the Self-Similarity Method

Akın

2022

Axioms

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In this study, an interactive Ising model having the nearest and prolonged next-nearest neighbors defined on a Cayley tree is considered. Inspired by the results obtained for the one-dimensional Ising model, we will construct the partition function and then calculate the free energy of the Ising model having the prolonged next nearest and nearest neighbor interactions and external field on a two-order Cayley tree using the self-similarity of the semi-infinite Cayley tree. The phase transition problem for the Ising system is investigated under the given conditions.

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

Cited by 11 publications

References 42 publications

Determination of paramagnetic and ferromagnetic phases of an Ising model on a third-order Cayley tree

Determination of paramagnetic and ferromagnetic phases of an Ising model on a third-order Cayley tree

Phase diagrams of lattice models on Cayley tree and chandelier network: a review

Calculation of the Free Energy of the Ising Model on a Cayley Tree via the Self-Similarity Method

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