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

doi:10.3390/math9070776

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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, Osamu 1; Zhang, Zhidong 2
刊名	MATHEMATICS
	2021-04-01
卷号	9 期号:7 页码:28
关键词	ferromagnetic 3D Ising model Clifford algebra Riemann– Hilbert problem trivialization of topological structure monoidal transformation
DOI	10.3390/math9070776
通讯作者	Zhang, Zhidong(zdzhang@imr.ac.cn)
英文摘要	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.
资助项目	National Natural Science Foundation of China
WOS研究方向	Mathematics
语种	英语
出版者	MDPI
WOS记录号	WOS:000638708700001
资助机构	National Natural Science Foundation of China
内容类型	期刊论文
源URL	[http://ir.imr.ac.cn/handle/321006/162343]
专题	金属研究所_中国科学院金属研究所
通讯作者	Zhang, Zhidong
作者单位	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
推荐引用方式 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.