On Quantum Markov Chains on Cayley Tree I: Uniqueness of the Associated Chain With Xy-Model on the Cayley Tree of Order Two

Abstract

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 conditions, and define QMC as a weak limit of those states which depends on the boundary conditions. Using the provided construction we investigate QMC associated with XY -model on a Caylay tree of order two. We prove uniqueness of QMC associated with such a model, this means the QMC does not depend on the boundary conditions. Mathematics Subject Classification: 46L53, 60J99, 46L60, 60G50, 82B10, 81Q10, 94A17.

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

“…If a tree is not one-dimensional lattice, then it is expected (from a physical point of view) the existence of a phase transition for quantum Markov chains constructed over such a tree. In [10] we have provided a construction of forward QMC, such states are different from backward QMC. In that construction, a QMC is defined as a weak limit of finite volume states with boundary conditions.…”

Phase Transitions for Quantum Markov Chains Associated with Ising Type Models on a Cayley Tree

“…The projectivity of W n] yields the equality (5) for ϕ (n) w 0 ,h , therefore, from (21) we conclude that ϕ w 0 ,h is a forward QMC. ,(x,i+1)) = 1 I for all x ∈ L, then we get a QMC constructed in [11]. Therefore, the provided construction extensions ones given in [11,12].…”

Phase transitions for XY-model on the Cayley tree of order three in quantum Markov chain scheme

“…Namely, we have [4]). Hence, the main result of the present paper totally differs from [4], and shows by increasing the dimension of the tree we are getting the phase transition.…”

“…Investigating the dynamical system (4) we prove that if β ∈ (0, β * ] ∪ [β * , ∞) then Eqs. (1) have only the following…”