On Translation Invariant Quantum Markov Chains associated with Ising-XY models on a Cayley tree

37th International Conference on Quantum Probability and Related Topics

Mukhamedov, Farrukh^1 ; Barhoumi, Abdessatar^2 ; Souissi, Abdessatar^3 ; El Gheteb, Soueidy^4

Department of Mathematical Sciences, College of Science, United Arab Emirates University, P.O. Box, Al Ain, Abu Dhabi

15551, United Arab Emirates^1

Department of Mathematics, Nabeul Preparatory Engineering Institute, Campus Universitairy, Carthage University, Mrezgua, Nabeul

8000, Tunisia^2

Department of Mathematics, Marsa Preparatory Institute for Scientific and Technical Studies, Carthage University, Tunisia^3

Department of Mathematics, Carthage University, Tunisia^4

关键词: Cayley trees;    Competing interactions;    Non-commutative;    Translation invariants;    XY interaction;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/819/1/012006/pdf
DOI  :  10.1088/1742-6596/819/1/012006

来源: IOP

【 摘 要 】

In this paper, we consider the Ising-XY model with competing interactions on the Cayley tree of order two. This model can be seen as a non-commutative (i.e. J-XY -interactions on next-neighbor vertices) perturbation of the classical Ising model on the Cayley tree. For the considered model we establish the existence of three translation-invariant quantum Markov chains. We notice that if the XY -interactions vanish, i.e. J = 0, then one gets the Ising model. If the classical Ising model vanishes in the considered model, then we obtain XY -model for which it turns out there exists only one translation invariant QMC.