A Class of Quantum Markov Fields on Tree-like Graphs: Ising-type Model on a Husimi Tree

A new class of forward quantum Markov fields (FQMFs) is introduced. The structure of these quantum Markov fields is investigated in the finer structure of a quasi-local algebra of observable over a tree-like graph. We provide an effective construction of a class of FQMCs. Moreover, we show the existence of three FMRFs associated with an Ising type model on a Husimi tree.

Strange Attractors in an Antiferromagnetic Ising Model

The three-site antiferromagnetic Ising model on Husimi tree is investigated in an external magnetic field. The full bifurcation diagram, including chaos, of the magnetization is exhibited. With the "thermodynamic formalism", we investigate the antiferromagnetic Ising model in the case of fully developed chaos and describe the chaotic properties of this statistical mechanical system via the invariants characterizing a strange attractor. It is shown that this system displays in the chaotic region a phase transition at a positive "temperature" whereas in a class of maps close to x→ 4x(1-x), the phase transitions occur at negative "temperatures". The Frobenius-Perron recursion equation is numerically solved and the density of the invariant measure is obtained.