TY - JOUR

T1 - On an Algebraic Property of the Disordered Phase of the Ising Model with Competing Interactions on a Cayley Tree

AU - Mukhamedov, Farrukh

AU - Barhoumi, Abdessatar

AU - Souissi, Abdessatar

N1 - Publisher Copyright:
© 2016, Springer Science+Business Media Dordrecht.

PY - 2016/12/1

Y1 - 2016/12/1

N2 - It is known that the disordered phase of the classical Ising model on the Caley tree is extreme in some region of the temperature. If one considers the Ising model with competing interactions on the same tree, then about the extremity of the disordered phase there is no any information. In the present paper, we first aiming to analyze the correspondence between Gibbs measures and QMC’s on trees. Namely, we establish that states associated with translation invariant Gibbs measures of the model can be seen as diagonal quantum Markov chains on some quasi local algebra. Then as an application of the established correspondence, we study some algebraic property of the disordered phase of the Ising model with competing interactions on the Cayley tree of order two. More exactly, we prove that a state corresponding to the disordered phase is not quasi-equivalent to other states associated with translation invariant Gibbs measures. This result shows how the translation invariant states relate to each other, which is even a new phenomena in the classical setting. To establish the main result we basically employ methods of quantum Markov chains.

AB - It is known that the disordered phase of the classical Ising model on the Caley tree is extreme in some region of the temperature. If one considers the Ising model with competing interactions on the same tree, then about the extremity of the disordered phase there is no any information. In the present paper, we first aiming to analyze the correspondence between Gibbs measures and QMC’s on trees. Namely, we establish that states associated with translation invariant Gibbs measures of the model can be seen as diagonal quantum Markov chains on some quasi local algebra. Then as an application of the established correspondence, we study some algebraic property of the disordered phase of the Ising model with competing interactions on the Cayley tree of order two. More exactly, we prove that a state corresponding to the disordered phase is not quasi-equivalent to other states associated with translation invariant Gibbs measures. This result shows how the translation invariant states relate to each other, which is even a new phenomena in the classical setting. To establish the main result we basically employ methods of quantum Markov chains.

KW - Cayley tree

KW - Competing interaction

KW - Disordered phase

KW - Ising type model

KW - Quantum Markov chain

KW - Quasi-equivalence

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U2 - 10.1007/s11040-016-9225-x

DO - 10.1007/s11040-016-9225-x

M3 - Article

AN - SCOPUS:84990180833

SN - 1385-0172

VL - 19

JO - Mathematical Physics Analysis and Geometry

JF - Mathematical Physics Analysis and Geometry

IS - 4

M1 - 21

ER -