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An approach to models of order-disorder and Ising lattices. (English) Zbl 1202.82020

Summary: The present paper develops an approach to the famous problem presented by L. Onsager [Phys. Rev., II. Ser. 65, 117–149 (1944; Zbl 0060.46001)] and its further investigation proposed in a recent work by Z.-D. Zhang [Philosophical Magazine 87, No. 34, 5309–5419 (2007)]. The above works give quaternion-based two- and three-dimensional (quantum) models of order-disorder transition and simple orthorhombic Ising lattices. The general methods applied by Zhang refer to opening knots by a rotation in a higher dimensional space, introduction of weight factor (his Conjecture 1 and 2) and important commutators.
The main objective of the present paper is to reformulate the algebraic part of the theory in terms of a quaternionic sequence of Jordan algebras and to look at some of the geometrical aspects of simple orthorhombic Ising-Onsager-Zhang lattices. The relationship with Bethe-type fractals, Kikuchi-type fractals, and fractals of the algebraic structure and, moreover, the duality for fractal sets and lattice models on fractal sets is discussed. A simple description in terms of fractals corresponding to algebraic structure involving a quaternionic sequence (\({\mathcal H}^4_q\)) of Jordan algebras appears to be possible. Physically, models of (\({\mathcal H}^4_q\)) for \(q = 5 \cdot 2^{2} = 20\) for the melting, \(q = 9 \cdot 2^{6} = 576\) for binary alloys, and \(q = 13 \cdot 2^{10} = 13 312\) for ternary alloys are obtained.

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
28A80 Fractals
17C90 Applications of Jordan algebras to physics, etc.

Citations:

Zbl 0060.46001
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