A Method of Riemann-Hilbert Problem for Zhang's Conjecture 1 in a Ferromagnetic 3D Ising Model: Trivialization of Topological Structure | |
Suzuki, Osamu1; Zhang, Zhidong2 | |
Corresponding Author | Zhang, Zhidong(zdzhang@imr.ac.cn) |
2021-04-01 | |
Source Publication | MATHEMATICS |
Volume | 9Issue:7Pages:28 |
Abstract | 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. |
Keyword | ferromagnetic 3D Ising model Clifford algebra Riemann– Hilbert problem trivialization of topological structure monoidal transformation |
Funding Organization | National Natural Science Foundation of China |
DOI | 10.3390/math9070776 |
Indexed By | SCI |
Language | 英语 |
Funding Project | National Natural Science Foundation of China |
WOS Research Area | Mathematics |
WOS Subject | Mathematics |
WOS ID | WOS:000638708700001 |
Publisher | MDPI |
Citation statistics | |
Document Type | 期刊论文 |
Identifier | http://ir.imr.ac.cn/handle/321006/162344 |
Collection | 中国科学院金属研究所 |
Corresponding Author | Zhang, Zhidong |
Affiliation | 1.Nihon Univ, Dept Comp & Syst Anal, Coll Humanities & Sci, Setagaya Ku, Sakurajosui 3-25-40, Tokyo 1568550, Japan 2.Chinese Acad Sci, Shenyang Natl Lab Mat Sci, Inst Met Res, 72 Wenhua Rd, Shenyang 110016, Peoples R China |
Recommended Citation GB/T 7714 | Suzuki, Osamu,Zhang, Zhidong. A Method of Riemann-Hilbert Problem for Zhang's Conjecture 1 in a Ferromagnetic 3D Ising Model: Trivialization of Topological Structure[J]. MATHEMATICS,2021,9(7):28. |
APA | Suzuki, Osamu,&Zhang, Zhidong.(2021).A Method of Riemann-Hilbert Problem for Zhang's Conjecture 1 in a Ferromagnetic 3D Ising Model: Trivialization of Topological Structure.MATHEMATICS,9(7),28. |
MLA | Suzuki, Osamu,et al."A Method of Riemann-Hilbert Problem for Zhang's Conjecture 1 in a Ferromagnetic 3D Ising Model: Trivialization of Topological Structure".MATHEMATICS 9.7(2021):28. |
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