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

Osamu Suzuki and Zhidong Zhang
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Osamu Suzuki: Department of Computer and System Analysis, College of Humanities and Sciences, Nihon University, Sakurajosui 3-25-40, Setagaya-ku, Tokyo 156-8550, Japan
Zhidong Zhang: Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110016, China

Mathematics, 2021, vol. 9, issue 7, 1-28

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.

Keywords: ferromagnetic 3D Ising model; Clifford algebra; Riemann–Hilbert problem; trivialization of topological structure; monoidal transformation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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