On an Algebraic Property of the Disordered Phase of the Ising Model with Competing Interactions on a Cayley Tree
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.
KeywordsQuantum Markov chain Cayley tree Ising type model Competing interaction Quasi-equivalence Disordered phase
Mathematics Subject Classifications (2010)46L53 60J99 46L60 60G50 82B10 81Q10 94A17
- 5.Accardi, L., Fidaleo, F.: On the structure of quantum Markov fields. In: Freudenberg, W. (ed.) Proceedings Burg Conference 15–20 March 2001. QP–PQ Series 15, pp. 1–20. World Scientific, Singapore (2003)Google Scholar
- 19.Georgi, H.-O.: Gibbs Measures and Phase Transitions, de Gruyter Studies in Math, vol. 9. Walter de Gruyter, Berlin (1988)Google Scholar
- 24.Mossel, E.: Peres Y. Information flow on trees. In: Graphs, Morphisms and Statistical Physics, pp. 155–170. AMS (2004)Google Scholar
- 31.Ostilli, M.: Cayley Trees and Bethe Lattices: A concise analysis for mathematicians and physicists. Physica A 391, 3417–3423 (2012)Google Scholar