An Approach to Models of Order-Disorder and Ising Lattices

Abstract

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 (H(q)(4)) of P. Jordan's algebras appears to be possible. Physically we obtain models of (H(q)(4)) 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

On the Ternary Approach to Clifford Structures and Ising Lattices

“…We conclude with a study of a composition algebra and self-dual perfect JNW-systems. Continuing our previous papers [10,11,13] we prove in this case it is one of quaternion and octonion algebras, and this needs a study of the product table of sedenion algebra. The idea of applying the JNW-systems, considered in Section 3 and thereafter, to binary and ternary alloys may be to some extent illustrated for instance by the first step of fractal construction related to an AB 3 binary alloy of an fcc lattice and (111) surface orientation [2, Fig.…”

“…The model consisting of an orthombic lattice constructed by m rows and n sites per row in one of the planes involves [11] 2 n· -dimensional matrices s α r,s , α = 1, 2, 3, 1 ≤ s ≤ , 1 ≤ r ≤ n, related to the so-called split-quaternions and, correspondingly, to the familiar Pauli matrices…”

“…As in [11], we construct a base system of ternaries Z(n) and generate the total system. For n = 3 we take (e 1 , e 2 , e 3 ) and let Z(3) = (1, 2, 3 The relation * 1 , connected with the vertex e 1 , is visualized in Fig.…”

“…In the case when A is an associative, it becomes a Clifford algebra when it additionally satisfies the following condition: e 2 j = η j e i for some i and η j ∈ R. The following theorem has been proven in [11]:…”

Proposed connection between critical exponents and fractal dimensions in the Ising model

“…Although the Ising system considered here is not a truly dynamic one, analogous to the definition of the fractal dimensions in other systems, it seems reasonable for us to consider that physical quantities near a critical point could be measured by some parameters (i.e., temperature and a magnetic field) so that the fractal dimensions could be defined also for the critical phenomena. One of us (ZDZ [6]; see also [7] and [8]) has, by means of two conjectures, obtained precise proposals for the six critical exponents of the Ising model Hamiltonian in three dimensions (d = 3). Using previously known results for d = 1, 2 and 4, Table 1 has thereby been constructed.…”