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 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 Cited Times:1[WOS]   [WOS Record]     [Related Records in WOS] 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, Japan2.Chinese Acad Sci, Shenyang Natl Lab Mat Sci, Inst Met Res, 72 Wenhua Rd, Shenyang 110016, Peoples R China Recommended CitationGB/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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