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

Zhang, Zhidong

doi:10.1007/978-3-319-72374-7_27

Cited by 2 publications

(2 citation statements)

References 52 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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“…After the publication of Zhang-Suzuki-March's work [8], up to date, no further criticisms have been published. It was suggested that the approximation techniques can be utilized to obtain the non-locality part of the partition function (as well as the thermodynamic physical properties) by extracting the approximation values from the exact solution [54,55].…”

Section: Introductionmentioning

confidence: 99%

A Method of Riemann–Hilbert Problem for Zhang’s Conjecture 1 in a Ferromagnetic 3D Ising Model: Trivialization of Topological Structure

Сузукі

Zhang

2021

Mathematics

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A method of the Riemann–Hilbert problem is applied for Zhang’s conjecture 1 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in the zero external field and the solution to the Zhang’s conjecture 1 is constructed by use of the monoidal transform. At first, the knot structure of the ferromagnetic 3D Ising model in the zero external field is determined and the non-local behavior of the ferromagnetic 3D Ising model can be described by the non-trivial knot structure. A representation from the knot space to the Clifford algebra of exponential type is constructed, and the partition function of the ferromagnetic 3D Ising model in the zero external field can be obtained by this representation (Theorem I). After a realization of the knots on a Riemann surface of hyperelliptic type, the monodromy representation is realized from the representation. The Riemann–Hilbert problem is formulated and the solution is obtained (Theorem II). Finally, the monoidal transformation is introduced for the solution and the trivialization of the representation is constructed (Theorem III). By this, we can obtain the desired solution to the Zhang’s conjecture 1 (Main Theorem). The present work not only proves the Zhang’s conjecture 1, but also shows that the 3D Ising model is a good platform for studying in deep the mathematical structure of a physical many-body interacting spin system and the connections between algebra, topology, and geometry.

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

Cited by 2 publications

References 52 publications

Clifford algebra approach of 3D Ising model

Clifford algebra approach of 3D Ising model

A Method of Riemann–Hilbert Problem for Zhang’s Conjecture 1 in a Ferromagnetic 3D Ising Model: Trivialization of Topological Structure

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