An Approach to Models of Order-Disorder and Ising Lattices

• Published in: Advances in applied Clifford algebras Vol. 20; no. 3-4; pp. 733 - 743
• Main Authors: Ławrynowicz, J., Marchiafava, S., Niemczynowicz, A.
• Format: Journal Article
• Language: English
• Published: Basel Birkhäuser-Verlag 01-10-2010
• Subjects: Applications of Mathematics; Mathematical and Computational Physics; Mathematical Methods in Physics; Physics; Physics and Astronomy; Theoretical; Secondary 82D25, 81R05, 15A66; quaternion; Primary 82C44; crystal; Ising lattice; Clifford algebra; Jordan algebra; disordered system
• ISSN: 0188-7009; 1661-4909
• DOI: 10.1007/s00006-010-0219-7

.
The present paper develops the approach to the famous problem presented by L. Onsager (1944) and its further investigation proposed in a recent work by Z.-D. Zhang (2007). The above works give quaternion-based two- and three-dimensional (quantum) models of order-disorder transition and simple orthorhombic Ising lattices (1925). 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 the quaternionic sequence of Jordan algebras and to look at some of the geometrical aspects of simple orthorhombic Ising-Onsager-Zhang lattices. The present authors discuss 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. A simple description in terms of fractals corresponding to algebraic structure involving the quaternionic sequence (
) of P. Jordan’s algebras appears to be possible. Physically we obtain models of (
) for
q
= 5 · 2
2
= 20 for the melting,
q
= 9 · 2
6
= 576 for binary alloys, and
q
= 13 · 2
10
= 13 312 for ternary alloys.