On Ising model with four competing interactions on Cayley tree
type: Item
instance of: scholarly article
title: On Ising model with four competing interactions on Cayley tree
zbMATH Open document ID: 1185.60112
published in: Mathematical Physics, Analysis and Geometry
publication date: 2009-07-10T00:00:00Z
review text: The paper studies a spin model on a Cayley tree of order two. Its Hamiltonian \(H\) includes four competing interactions (external field, nearest neighbor, second neighbors and triples of neighbors). For a sequence \(V_1 \subset V_2 \subset \dots\) with \(\bigcup_{n=1}^{\infty}V_n=V\) and Gibbs measures \(\mu^{(n)}\) on \(\{-1,+1\}^{V_n}\) (\(n\in\mathbb{N}\)) satisfying the consistency conditions one obtains a limit Gibbs measure \(\mu\) on \(\{-1,+1\}^V\) as \(n\to \infty\). The criterion (see equation (5) of the paper) to meet these conditions is given in terms of the model parameters appearing in \(H\) and in the description of measures \(\mu^{(n)}\). In a particular case it is proved that there are three translation-invariant Gibbs measures \(\mu_1,\mu_2,\mu_3\) as equation (5) admits different solutions (i.e. there is a phase transition). The complete description of the periodic Gibbs measures for the model is provided as well. Moreover, the continuum of distinct non-periodic extreme Gibbs measures is constructed.
reviewed by: Alexander V. Bulinski
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B20
Mathematics Subject Classification ID: 82B05
zbMATH DE Number: 5577254
zbMATH Keywords: Cayley tree
zbMATH Keywords: Ising model
zbMATH Keywords: competing interaction
zbMATH Keywords: phase transition
zbMATH Keywords: Gibbs measure
author: Nasir N. Ganikhodjaev
author: U. A. Rozikov
MaRDI profile type: Publication
full work available at URL: s11040-009-9056-0
OpenAlex ID: W2080845852
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DOI: 10.1007/S11040-009-9056-0
description: scientific article; zbMATH DE number 5577254