On the Ternary Approach to Clifford Structures and Ising Lattices

Ławrynowicz, Julian; Сузукі, Осаму; Niemczynowicz, Agnieszka

doi:10.1007/s00006-012-0360-6

Cited by 13 publications

(16 citation statements)

References 12 publications

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“…As pointed out in ref. [20][21][22], one of the present authors (ZDZ) gave quaternion-based 3D (quantum) models of order-disorder transition for simple orthorhombic Ising lattices, based on Jordan-von Neumann-Wigner procedure [14]. Two matrices A and B, which are in Jordan algebra [12][13][14]., one has:…”

Section: Proofmentioning

confidence: 99%

Clifford algebra approach of 3D Ising model

Zhang

Сузукі

March

2018

Adv. Appl. Clifford Algebras

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We develop a Clifford algebra approach for 3D Ising model. We first note the main difficulties of the problem for solving exactly the model and then emphasize two important principles (i.e., Symmetry Principle and Largest Eigenvalue Principle) that will be used for guiding the path to the desired solution. By utilizing somein Jordan algebras is applied instead of the usual matrix multiplication AB (Theorem IV: Commutation Theorem).. This can be realized by time-averaging t systems of the 3D Ising models with time evaluation. In order to determine the rotation angle for the local transformation, the star-triangle relationship of the 3D Ising model is employed for Curie temperature, which is the solution of generalized Yang-Baxter equations in the continuous limit. Finally, the topological phases generated on the eigenvectors are determined, based on the relation with the Ising gauge lattice theory. ..

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Section: Proofmentioning

confidence: 99%

Clifford algebra approach of 3D Ising model

Zhang

Сузукі

March

2018

Adv. Appl. Clifford Algebras

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“…(11) and (12)) can be negligible, so matrices V 1 , V 2 and V 3 in eqs. (11), (12) and (13) can be reduced to: [6] } ' ' exp{ } ' ' exp{…”

Section: Hamiltonian Transfer Matrixes and Consequences Of The Cmentioning

confidence: 99%

“…[2] can be made more elegant and simple by the use of Clifford structures and the P. Jordan structures. [9][10][11][12] The natural appearance of the multiplication ( )…”

Section: A Algebraic Aspects and Jordan-von Neumann-wigner Proceduresmentioning

confidence: 99%

“…[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.…”

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confidence: 99%

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

Zhang¹

2013

Chinese Phys. B

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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...

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Topological Effects and Critical Phenomena in the Three-Dimensional (3D) Ising Model

Zhang

2018

Many-Body Approaches at Different Scales

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On the Ternary Approach to Clifford Structures and Ising Lattices

Cited by 13 publications

References 12 publications

Clifford algebra approach of 3D Ising model

Clifford algebra approach of 3D Ising model

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

Topological Effects and Critical Phenomena in the Three-Dimensional (3D) Ising Model

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