Narrow Results

In this paper we consider nearest neighbour models where the spin takes values in the set Φ = {η₁, η₂, …, η_q} and is assigned to the vertices of the Cayley tree Γ^k. The Hamiltonian is defined by some given λ-function. We find a condition for the function λ to determine the type of the von Neumann algebra generated by the GNS - construction associated with the quantum Markov state corresponding to the unordered phase of the λ-model. Also we give some physical applications of the obtained result.

In the present paper we continue our investigations started in [Accardi L., Ohno, H., Mukhamedov, F., Quantum Markov fields on graphs, Inf. Dim. Analysis, Quantum Probab. Related Topics (accepted) arxiv: 0911.1667]. In [Accardi L., Mukhamedov, F., Saburov M. On Quantum Markov chains on Cayley tree and associated chains with XY-model arxiv: 1004.3623] we provided a construction of forward and backward Quantum Markov Chains (QMC) defined on the Cayley tree, and established uniqueness of QMC associated with XY-model on a Cayley tree order 2. In the present paper we study the same model on a Cayley tree order 3. Surprisingly in this case, we establish a phase transition (i.e. existence of two distinct quantum Markov chains) for the considered model on the Cayley tree order 3.

In the present paper we study forward Quantum Markov Chains (QMC) associated with Ising model on Cayley tree of order k. 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 Ising model on a Cayley tree of order k in QMC scheme. By the phase transition we mean the existence of two not quasi equivalent QMC for the given family of interaction operators.