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Infinite Dimensional Analysis Quantum Probability and Related Topics

Quantum Ising model with generalized competingXY-interactions on a Cayley tree

Farrukh Maksutovich Mukhamedov, El Gheteb Soueidi

Published Apr 20, 2022• 1 citation

Abstract

In this paper, the phase transition phenomena for the Ising model (with nearest-neighbor interaction [Formula: see text]) but with quantum generalized competing [Formula: see text]-interactions ([Formula: see text] and [Formula: see text] coupling constants) are treated by means of a quantum Markov chain (QMC) approach. We point out that the case [Formula: see text] has been carried out in Ref. 32. Note that if [Formula: see text], then it turns out that phase transition exists, for any value of [Formula: see text], while the Ising coupling constant should satisfy [Formula: see text]. This means that the Ising interaction is dominated in the considered situation, i.e. the X-competing interactions’ role is negligible. This kind of phenomena was not detected in this mentioned paper. Phase transition means the existence of at least two distinct QMCs which are not quasi-equivalent and their supports do not overlap. To prove the quasi-equivalence, it is first established that the QMCs satisfy clustering property.

DOI: 10.1142/s0219025722500102

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In this paper, the phase transition phenomena for the Ising model (with nearest-neighbor interaction [Formula: see text]) but with quantum generalized competing [Formula: see text]-interactions ([Formula: see text] and [Formula: see text] coupling constants) are treated by means of a quantum Markov chain (QMC) approach. We point out that the case [Formula: see text] has been carried out in Ref. 32. Note that if [Formula: see text], then it turns out that phase transition exists, for any value of [Formula: see text], while the Ising coupling constant should satisfy [Formula: see text]. This means that the Ising interaction is dominated in the considered situation, i.e. the X-competing interactions’ role is negligible. This kind of phenomena was not detected in this mentioned paper. Phase transition means the existence of at least two distinct QMCs which are not quasi-equivalent and their supports do not overlap. To prove the quasi-equivalence, it is first established that the QMCs satisfy clustering property.

The main aim of the present paper by means of the quantum Markov chain (QMC) approach is to establish the existence of a phase transition for the quantum Ising model with competing XY interaction. In this scheme, the C*-algebraic approach is employed to the phase transition problem. Note that these kinds of models do not have one-dimensional analogs, 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 QMCs that are not quasi-equivalent and their supports do not overlap. Moreover, it is also shown that the QMC associated with the model has a clustering property, which implies that the von Neumann algebras corresponding to the states are factors.

In this work, we construct a quantum Markov chain (QMC) associated by the classical Ising model with competing interactions on the Cayley tree of order two. In the construction QMC is defined as a weak limit of finite volume states on quasi-local algebras with boundary conditions. We point out that phase transitions in a quantum setting play an important role to understand quantum spin systems We have defined a notion of phase transition in QMC scheme. Namely, such a notion is based on the quasi-equivalence of QMC. Therefore, such a phase transition is purely non-commutative. In this work we establish 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.

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.

The main aim of the present paper is to provide a new construction of quantum Markov chain (QMC) on arbitrary order Cayley tree. 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. Note that this construction reminds statistical mechanics models with competing interactions on trees. If one considers one dimensional tree, then the provided construction reduces to well-known one, which was studied by the first author. Our construction will allow to investigate phase transition problem in a quantum setting.