Response to ‘Comment on a recent conjectured solution of the three-dimensional Ising model’

Zhang, Z. D.

doi:10.1080/14786430802537720

Cited by 16 publications

(40 citation statements)

References 24 publications

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“…There have been several rounds of comments/responses between some statistical physicists and Zhang [30][31][32]39,40,46,47]. We do not want to repeat Zhang's responses published already [45][46][47][48][49][50]. Just mention briefly: The error for the Jordan-Wigner transform in Zhang's original paper [44] is not a problem, which has been corrected in ref.…”

Section: Introduction To 3d Ising Modelmentioning

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: Introduction To 3d Ising Modelmentioning

confidence: 99%

Clifford algebra approach of 3D Ising model

Zhang

Сузукі

March

2018

Adv. Appl. Clifford Algebras

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“…[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).…”

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confidence: 99%

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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“…The objects of type (3.1) appear as a result of calculations in [16] and [20], but they inspire in us a relationship with the two types of fractals mentioned.…”

Section: Ising-onsager-zhang Approachmentioning

confidence: 99%

“…instead of the usual matrix multiplication replaces in elegant way the desire of Ising [3], Onsager [16], Zhang [20], and Staples [18] of finding commutative subalgebras of the given algebra and studying their properties related to the superposition of bilinear forms.…”

Section: Jordan-von Neumann-wigner Proceduresmentioning

confidence: 99%

“…The models of Ising [3] and Onsager [16] are extended by Zhang [20] so that the resulting model applies to a simple orthorhombic crystal lattice in a threedimensional space. This extension involves, in particular, opening knots by a rotation in a higher dimensional space, and studying important commutators in the corresponding algebra.…”

Section: Ising-onsager-zhang Approachmentioning

confidence: 99%

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On the Ternary Approach to Clifford Structures and Ising Lattices

Ławrynowicz

Сузукі

Niemczynowicz

2012

Adv. Appl. Clifford Algebras

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Abstract. We continue to modify and simplify the Ising-Onsager-Zhang procedure for analyzing simple orthorhombic Ising lattices by considering some fractal structures in connection with Jordan and Clifford algebras and by following Jordan-von Neumann-Wigner (JNW) approach. We concentrate on duality of complete and perfect JNW-systems, in particular ternary systems, analyze algebras of complete JNW-systems, and prove that in the case of a composition algebra we have a self-dual perfect JNW-system related to quaternion or octonion algebras. In this context, we are interested in the product table of the sedenion algebra.Mathematics Subject Classification (2010). Primary 82C44; Secondary 82D25, 81R05, 15A66.

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Response to ‘Comment on a recent conjectured solution of the three-dimensional Ising model’

Cited by 16 publications

References 24 publications

Clifford algebra approach of 3D Ising model

Clifford algebra approach of 3D Ising model

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

On the Ternary Approach to Clifford Structures and Ising Lattices

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