Mathematical Outlook of Fractals and Chaos Related to Simple Orthorhombic Ising-Onsager-Zhang Lattices

Ławrynowicz, Julian; Marchiafava, S.; Nowak-Kȩpczyk, Małgorzata

doi:10.1142/9789814277723_0018

Cited by 8 publications

(10 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…1/2) Ising model on a 3D decorated lattice with a layered magnetic structure were investigated within the framework of a precise mapping relationship to the simple spin-1/2 Ising model on a tetragonal lattice by adopting Zhang's solution [8]. Very recently, Lawrynowicz et al [9] reformulated the algebraic part of the theory in terms of the quaternionic sequence of Jordan algebras to look at some of the geometrical aspects of simple orthorhombic Ising lattices, and represented a mathematical outlook of fractals and chaos related to these 3D Ising lattices [10]. These motivate us, in this study, to first compare the recent theoretical 3D Ising results [1] with more experimental data in various magnetic materials.…”

Section: Introductionmentioning

confidence: 99%

Three-dimensional (3D) Ising universality in magnets and critical indices at fluid–fluid phase transition

Zhang

March

2011

Phase Transitions

View full text Add to dashboard Cite

Section: Introductionmentioning

confidence: 99%

Three-dimensional (3D) Ising universality in magnets and critical indices at fluid–fluid phase transition

Zhang

March

2011

Phase Transitions

View full text Add to dashboard Cite

“…[2] does not affect the validity of the putative exact solution, since the conjectures serve for solving the topological problems existing in that equation.Further progresses have been made in ref. [4,7,[9][10][11][12] . In ref.…”

mentioning

confidence: 99%

“…[2] was reformulated in terms of the quaternionic sequence of Jordan algebras to look at the geometrical aspects of simple orthorhombic Ising lattices, [9] and fractals and chaos related to these 3D Ising lattices were investigated. [10][11][12] In this work, we represent an overview of the mathematical structure of the 3D Ising model from the viewpoints of topologic, algebraic and geometric aspects,* and attempt to bridge the gaps in the conjectures. This paper is arranged as follows: In Section II, the Hamiltonian, transfer matrixes, boundary conditions, and the consequences of the two conjectures are introduced briefly (and the technical errors in Eqs.…”

mentioning

confidence: 99%

Mathematical structure of the three-dimensional (3D) Ising model

Zhang¹

2013

Chinese Phys. B

View full text Add to dashboard Cite

An overview of the mathematical structure of the three-dimensional (3D) Ising model is given, from the viewpoints of topologic, algebraic and geometric aspects. By analyzing the relations among transfer matrices of the 3D Ising model, Reidemeister moves in the knot theory, Yang-Baxter and tetrahedron equations, the following facts are illustrated for the 3D Ising model: 1) The complexified quaternion basis constructed for the 3D Ising model represents naturally the rotation in a (3 + 1)dimensional space-time, as a relativistic quantum statistical mechanics model, which is consistent with the 4-fold integrand of the partition function by taking the time average. 2) A unitary transformation with a matrix being a spin representation in 2 n⋅l⋅o -space corresponds to a rotation in 2n⋅l⋅o-space, which serves to smooth all the crossings in the transfer matrices and contributes as the non-trivial topologic part of the partition function of the 3D Ising model. 3) A tetrahedron relation would ensure the commutativity of the transfer matrices and the integrability of the 3D Ising model, and its existence is guaranteed also by the Jordan algebra and the Jordan-von Neumann-Wigner procedures. 4) The unitary transformation for smoothing the crossings in the transfer matrices changes the wave functions by complex phases φ x , φ y , and φ z . The relation with quantum field and gauge theories, physical significance of weight factors are discussed in details. The conjectured exact solution is compared with numerical results, and singularities at/near infinite temperature are inspected.The analyticity in β = 1/(k B T) of both the hard-core and Ising models has been proved for β > 0, not for β = 0. Thus the high-temperature series cannot serve as a standard for judging a putative exact solution of the 3D Ising model. 2 in the 4D space to the energy spectrum of the system.After publication of ref. [2] , two rounds of exchanges of Comments / Responses / Rejoinders appeared in 2008-2009. [3-8] The main objections of these Comments and Rejoinders [3,5,6,8] are summarized briefly as follows: The conjectured solution disagrees with low-temperature and high-temperature series, while the convergence of the high-temperature series has been rigorously proved. There are some problems with weight factors w y and w z (such as they are first defined to be real values, but actually can be complex, and the different weights are taken for infinite and finite temperature). There is a technical error in eq. (15) of ref. [2] for the application of the Jordan-Wigner transformation. The main rebuttals in my Responses [4,7] are also summarized briefly here: The objections in the Comments/Rejoinders [3,5,6,8] are limited to the outcome of the calculations and there were no comments on the topology-based approach underlying the derivation. All the well-known theorems for the convergence of the high-temperature series are proved only for β (= 1/k B T) > 0, not for infinite temperature (β = 0). Exactly infinite temperature has been never touched in these the...

show abstract

“…The proof is analogous to that of the corresponding theorem given in [4][5][6]. However, in order to understand two important differences:…”

Section: We Havementioning

confidence: 82%

Structure fractals and para-quaternionic geometry

Ławrynowicz¹,

Vaccaro²

2011

Annales UMCS, Mathematica

View full text Add to dashboard Cite

Abstract. It is well known that starting with real structure, the CayleyDickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions p = 2, 4 and 8, respectively, but the procedure fails for p = 16 in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are n = 2 7 . Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of the flower type related to complex and Pauli structures and, in relation to the iteration process p → p + 2 → p + 4 → . . . , they have constructed 2 4 -dimensional "bipetals" for p = 9 and 2 7 -dimensional "bisepals" for p = 13. The objects constructed appear to have an interesting property of periodicity related to the gradating function on the fractal diagonal interpreted as the "pistil" and a family of pairs of segments parallel to the diagonal and equidistant from it, interpreted as the "stamens". The first named author, M. Nowak-Kępczyk, and S. Marchiafava (2006, 2009a, b) gave an effective, explicit determination of the periods and expressed them in terms of complex and quaternionic structures, thus showing the quaternionic background of that periodicity. In contrast to earlier results, the fractal bundle flower structure, in particular petals, sepals, pistils, and stamens are not introduced ab initio; they are quoted a posteriori, when they are fully motivated. Physical concepts of dual and conjugate objects as well as of antiparticles led us to extend the periodicity theorem to structure fractals in para-quaternionic formulation, applying some results in this direction by the second named author. The paper is concluded by outlining some applications.2000 Mathematics Subject Classification. 81R25, 32L25, 53A50, 15A66.

show abstract

Mathematical Outlook of Fractals and Chaos Related to Simple Orthorhombic Ising-Onsager-Zhang Lattices

Cited by 8 publications

References 0 publications

Three-dimensional (3D) Ising universality in magnets and critical indices at fluid–fluid phase transition

Three-dimensional (3D) Ising universality in magnets and critical indices at fluid–fluid phase transition

Mathematical structure of the three-dimensional (3D) Ising model

Structure fractals and para-quaternionic geometry

Contact Info

Product

Resources

About