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

Akın, Hasan

doi:10.5488/cmp.24.13001

Cited by 8 publications

(5 citation statements)

References 43 publications

(95 reference statements)

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“…The most interesting finding here is that the phase transition occurs when both J and J p are negative (in the anti-ferromagnetic case). For the Ising model with the same Hamiltonian, no phase transition has occurred in the anti-ferromagnetic regimes in previous studies (see [16,22]).…”

Section: Discussionmentioning

confidence: 65%

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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Section: Discussionmentioning

confidence: 65%

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

Akın

2022

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“…Mech., 2022, 053204], using the Kolmogorov consistency theorem, we established the translation invariant splitting Gibbs measures (TISGMs) connected to the Ising model with mixed spin (1,1/2) on the second-order Cayley tree. We showed that the mixed spin Ising model comprises three TISGMs in both the ferromagnetic and antiferromagnetic regions, in contrast to the classic Ising model [ [101], Akın, H., Physica B, 2022, 645, 414221]. In this context, considering the theoretical results, detailed analysis of the transitions between phases will be carried out by drawing phase diagrams.…”

Section: Acknowledgementsmentioning

confidence: 98%

“…• On a Cayley tree of order three, Akın [101] has recently analytically analysed the recurrence equations of an Ising model with three competing interactions. He precisely described the Ising model's paramagnetic and ferromagnetic phases.…”

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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“…In Ref. [16], we proved that the (1,1/2)-MSIM on a CT of order two has three TIGMs in both the ferromagnetic and anti-ferromagnetic regions, while the Ising model having single-spin [18,19,20,21,22] does not have such GMs in the anti-ferromagnetic region. At the same time, for the Potts models [23] on the CT, the phase transition occurs only in the ferromagnetic regime.…”

Section: Introductionmentioning

confidence: 95%

“…When performing in both ferromagnetic and anti-ferromagnetic regimes, since |λ 3 (θ, k)| > 1, there are repelling fixed points. The repelling fixed points indicate that there exists more than one GM and phase transition happens, while the singlespin Ising model does not have the TIGMs in the anti-ferromagnetic regime [18,20,21,33,37,38].…”

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confidence: 97%

The Extremality of Disordered Phases for the Mixed Spin-(1,1/2) Ising Model on Cayley Tree of Arbitrary Order

Akın

Mukhamedov

2023

Preprint

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The aim of this paper is to continue the investigation into the set of translation-invariant splitting Gibbs measures (TISGMs) for Ising model having the mixed spin (1,1/2) (shortly, (1,1/2)-MSIM) on a Cayley tree of arbitrary order. In our previous work [Akın and Mukhamedov, J. Stat. Mech. (2022) 053204], we provided a thorough explanation of the TISGMs, and studied the extremality of disordered phases using a Markov chain with a tree index on a semi-finite Cayley tree with order two. In this paper, we construct the TISGMs and tree-indexed Markov chains associated with to the model. Considering a tree-indexed Markov chain on a Cayley tree of any order, we clarify the extremality of the related disordered phases. By using the Kesten-Stigum condition (KSC), we investigate non-extremality of the disordered phases by means of the eigenvalues of the stochastic matrix associated with (1,1/2)-MSIM on a CT of order k≥2.

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

Cited by 8 publications

References 43 publications

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

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

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

The Extremality of Disordered Phases for the Mixed Spin-(1,1/2) Ising Model on Cayley Tree of Arbitrary Order

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