On Quantum Markov Chains on Cayley Tree II: Phase Transitions for the Associated Chain with XY-Model on the Cayley Tree of Order Three

@article{Accardi2010OnQM,
title={On Quantum Markov Chains on Cayley Tree II: Phase Transitions for the Associated Chain with XY-Model on the Cayley Tree of Order Three},
author={Luigi Accardi and Farrukh Mukhamedov and Mansoor Saburov},
journal={Annales Henri Poincar{\'e}},
year={2010},
volume={12},
pages={1109-1144}
}
• Published 10 November 2010
• Mathematics
• Annales Henri Poincaré
In the present paper, we study forward quantum Markov chains (QMC) defined on a Cayley tree. Using the tree structure of graphs, we give a construction of quantum Markov chains on a Cayley tree. By means of such constructions we prove the existence of a phase transition for the XY-model on a Cayley tree of order three in QMC scheme. By the phase transition we mean the existence of two distinct QMC for the given family of interaction operators $${\{K_{\langle x,y\rangle}\}}$$.
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References

SHOWING 1-10 OF 46 REFERENCES

• Mathematics
• 2010
In the present paper we study forward Quantum Markov Chains (QMC) defined on Cayley tree. A construction of such QMC is provided, namely we construct states on finite volumes with boundary
• Mathematics
• 2009
We introduce generalized quantum Markov states and generalized d-Markov chains which extend the notion quantum Markov chains on spin systems to that on $C^*$-algebras defined by general graphs. As
• Mathematics
• 2005
Motivated by the problem of finding a satisfactory quantum generalization of the classical random walks, we construct a new class of quantum Markov chains which are at the same time purely generated
Heuristically a quantum Markov chain indexed by N is a state ϕ on ⊗NB (B − a matrix algebra) whose sequence of density matrices has a local structure (corresponding to the physical intuition that the
A new approach to quantum Markov chains is presented. We first define a transition operation matrix (TOM) as a matrix whose entries are completely positive maps whose column sums form a quantum
• Mathematics
• 2007
We introduce the notion of Markov states and chains on the Canonical Anticommutation Relations algebra over ℤ, emphasizing some remarkable differences with the infinite tensor product case. We
• Mathematics
• 1999
We characterize a class of quantum Markov states in terms of a locality property of their modular automorphism group or, equivalently, of their φ-conditional expectations and we give an explicit
• Mathematics
• 2004
In the present paper the Ising model with competing binary (J) and binary (J1) interactions with spin values ±1, on a Cayley tree of order 2 is considered. The structure of Gibbs measures for the
• Computer Science, Mathematics
• 2007
It is shown that for tri-partite quantum states the quantum conditional information is always a lower bound for the minimum relative entropy distance to a quantum Markov chain state, but the distance can be much greater; indeed the two quantities can be of different asymptotic order and may even differ by a dimensional factor.