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

Abstract

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.Mathematics Subject Classification: 46L53, 60J99, 46L60, 60G50, 82B10, 81Q10, 94A17.

Quantum Markov states on Cayley trees

“…In section 7, we prove a reconstruction result. Finally, in section 8, we will establish that any locally faithful QMS associated with localized conditional expectations can be considered as Gibbs state corresponding to Hamiltonians (on the Cayley tree) with commuting competing interactions which implies that all QMC considered in [31,32] are indeed QMS.…”

“…its finer localization structure of the considered tree through considering suitable quasi-conditional expectation called localized, which keeps into account this finer localization and to prove the structure theorem corresponding to this localization. An interesting consequence of this structure theorem is that the notion of competing interactions, previously introduced by hands [31,32], now emerges as a consequence of the intrinsic denition combined with the structure theorem. Therefore, the present paper's main result differs from the non-homogeneous one dimensional case studied in [4].…”

“…Moreover, certain concrete examples were provided. This direction opened a new direction in the study and construction of QMC via investigation of lattice models on trees [9]- [12], [31]. Mostly, the existing works based on certain models over the Cayley trees (or Bathe lattices) [34].…”

“…On the other hand, in [37,38,45,46] a particular class of quantum Markov chains (QMC) associated to the Ising types models on the Cayley trees have been explored (see [17,29,53,55] for recent development on models over such trees). It turned out that the above considered QMCs fall to a special class called quantum Markov states (QMS) (see [42,44]).…”

“…First attempts regarding this goal have been done in [13], [12]. In this direction, quantum Markov chains on trees and their applications to quantum phase transition phenomena for concrete models were explored in an increasing number of works (see for instance [9,10,11], [37,38,39,40,41]).…”