On Quantum Markov Chains on Cayley Tree II: Phase Transitions for the Associated Chain with XY-Model on the Cayley Tree of Order Three (original) (raw)
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Phase transitions for XY-model on the Cayley tree of order three in quantum Markov chain scheme
Comptes Rendus Mathematique, 2011
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 <x,y> }.
On Quantum Markov Chains on Cayley tree and associated chains with XY-model
In this 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 Cayley 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.
On Quantum Markov Chains on Cayley tree II
2016
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 now quasi equivalent QMC for the given family of interaction operators {K <x,y> }.
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2011
In the present paper we study forward and backward Quantum Markov Chains (QMC) defined on Caylay tree. Constructions of such QMC are 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 provided constructions we investigated QMC associated with XY -model on a Caylay tree. We prove uniqueness of QMC associated with such a model, which 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 I
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Uniqueness of quantum Markov chains associated with an XY-model on a cayley tree of order 2
Mathematical Notes, 2011
We propose the construction of a quantum Markov chain that corresponds to a "forward" quantum Markov chain. In the given construction, the quantum Markov chain is defined as the limit of finite-dimensional states depending on the boundary conditions. A similar construction is widely used in the definition of Gibbs states in classical statistical mechanics. Using this construction, we study the quantum Markov chain associated with an XY -model on a Cayley tree. For this model, within the framework of the given construction, we prove the uniqueness of the quantum Markov chain i.e., we show that the state is independent of the boundary conditions.
On Quantum Markov Chains on Cayley tree III: Ising model
Journal of Statistical Physics
In this paper, we consider the classical Ising model on the Cayley tree of order k and show the existence of the phase transition in the following sense: there exists two quantum Markov states which are not quasi-equivalent. It turns out that the found critical temperature coincides with usual critical temperature.
Phase Transitions for Quantum Markov Chains Associated with Ising Type Models on a Cayley Tree
Journal of Statistical Physics, 2016
The main aim of the present paper is to prove the existence of a phase transition in quantum Markov chain (QMC) scheme for the Ising type models on a Cayley tree. Note that this kind of models do not have one-dimensional analogous, i.e. the considered model persists only on trees. In this paper, we provide a more general construction of forward QMC. In that construction, a QMC is defined as a weak limit of finite volume states with boundary conditions, i.e. QMC depends on the boundary conditions. Our main result states the existence of a phase transition for the Ising model with competing interactions on a Cayley tree of order two. By the phase transition we mean the existence of two distinct QMC which are not quasi-equivalent and their supports do not overlap. We also study some algebraic property of the disordered phase of the model, which is a new phenomena even in a classical setting.
Journal of Mathematical Physics
The main aim of the present paper is to establish the existence of a phase transition for the quantum Ising model with competing XY interactions within the quantum Markov chain (QMC) scheme. In this scheme, we employ the C *-algebraic approach to the phase transition problem. Note that these kinde of models do not have one-dimensional analogues, i.e. the considered model persists only on trees. It turns out that if the Ising part interactions vanish then the model with only competing XY-interactions on the Cayley tree of order two does not have a phase transition. By phase transition we mean the existence of two distinct QMC which are not quasi-equivalent and their supports do not overlap. Moreover, it is also shown that the QMC associated with the model have clustering property which implies that the von Neumann algebras corresponding to the states are factors.
Phase Transitions for quantum Ising model with competing XY -interactions on a Cayley tree
2019
The main aim of the present paper is to establish the existence of a phase transition for the quantum Ising model with competing XY interactions within the quantum Markov chain (QMC) scheme. In this scheme, we employ the C^*-algebraic approach to the phase transition problem. Note that these kinde of models do not have one-dimensional analogues, i.e. the considered model persists only on trees. It turns out that if the Ising part interactions vanish then the model with only competing XY -interactions on the Cayley tree of order two does not have a phase transition. By phase transition we mean the existence of two distinct QMC which are not quasi-equivalent and their supports do not overlap. Moreover, it is also shown that the QMC associated with the model have clustering property which implies that the von Neumann algebras corresponding to the states are factors.