A Method of the Riemann–Hilbert Problem for Zhang’s Conjecture 2 in a Ferromagnetic 3D Ising Model: Topological Phases

A method of the Riemann–Hilbert problem is employed for Zhang’s conjecture 2 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in a zero external magnetic field. In this work, we first prove that the 3D Ising model in the zero external magnetic field can b...

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Autores principales:	Zhidong Zhang, Osamu Suzuki
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	ferromagnetic 3D Ising model topological phase Gauss–Bonnet–Chern formula Riemann–Hilbert problem vertex operators vector bundle Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/15656cb6bb6441989ff0a35f6c3eb7fa
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spelling	oai:doaj.org-article:15656cb6bb6441989ff0a35f6c3eb7fa2021-11-25T18:17:19ZA Method of the Riemann–Hilbert Problem for Zhang’s Conjecture 2 in a Ferromagnetic 3D Ising Model: Topological Phases10.3390/math92229362227-7390https://doaj.org/article/15656cb6bb6441989ff0a35f6c3eb7fa2021-11-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/22/2936https://doaj.org/toc/2227-7390A method of the Riemann–Hilbert problem is employed for Zhang’s conjecture 2 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in a zero external magnetic field. In this work, we first prove that the 3D Ising model in the zero external magnetic field can be mapped to either a (3 + 1)-dimensional ((3 + 1)D) Ising spin lattice or a trivialized topological structure in the (3 + 1)D or four-dimensional (4D) space (Theorem 1). Following the procedures of realizing the representation of knots on the Riemann surface and formulating the Riemann–Hilbert problem in our preceding paper [O. Suzuki and Z.D. Zhang, Mathematics 9 (2021) 776], we introduce vertex operators of knot types and a flat vector bundle for the ferromagnetic 3D Ising model (Theorems 2 and 3). By applying the monoidal transforms to trivialize the knots/links in a 4D Riemann manifold and obtain new trivial knots, we proceed to renormalize the ferromagnetic 3D Ising model in the zero external magnetic field by use of the derivation of Gauss–Bonnet–Chern formula (Theorem 4). The ferromagnetic 3D Ising model with nontrivial topological structures can be realized as a trivial model on a nontrivial topological manifold. The topological phases generalized on wavevectors are determined by the Gauss–Bonnet–Chern formula, in consideration of the mathematical structure of the 3D Ising model. Hence we prove the Zhang’s conjecture 2 (main theorem). Finally, we utilize the ferromagnetic 3D Ising model as a platform for describing a sensible interplay between the physical properties of many-body interacting systems, algebra, topology, and geometry.Zhidong ZhangOsamu SuzukiMDPI AGarticleferromagnetic 3D Ising modeltopological phaseGauss–Bonnet–Chern formulaRiemann–Hilbert problemvertex operatorsvector bundleMathematicsQA1-939ENMathematics, Vol 9, Iss 2936, p 2936 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	ferromagnetic 3D Ising model topological phase Gauss–Bonnet–Chern formula Riemann–Hilbert problem vertex operators vector bundle Mathematics QA1-939
spellingShingle	ferromagnetic 3D Ising model topological phase Gauss–Bonnet–Chern formula Riemann–Hilbert problem vertex operators vector bundle Mathematics QA1-939 Zhidong Zhang Osamu Suzuki A Method of the Riemann–Hilbert Problem for Zhang’s Conjecture 2 in a Ferromagnetic 3D Ising Model: Topological Phases
description	A method of the Riemann–Hilbert problem is employed for Zhang’s conjecture 2 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in a zero external magnetic field. In this work, we first prove that the 3D Ising model in the zero external magnetic field can be mapped to either a (3 + 1)-dimensional ((3 + 1)D) Ising spin lattice or a trivialized topological structure in the (3 + 1)D or four-dimensional (4D) space (Theorem 1). Following the procedures of realizing the representation of knots on the Riemann surface and formulating the Riemann–Hilbert problem in our preceding paper [O. Suzuki and Z.D. Zhang, Mathematics 9 (2021) 776], we introduce vertex operators of knot types and a flat vector bundle for the ferromagnetic 3D Ising model (Theorems 2 and 3). By applying the monoidal transforms to trivialize the knots/links in a 4D Riemann manifold and obtain new trivial knots, we proceed to renormalize the ferromagnetic 3D Ising model in the zero external magnetic field by use of the derivation of Gauss–Bonnet–Chern formula (Theorem 4). The ferromagnetic 3D Ising model with nontrivial topological structures can be realized as a trivial model on a nontrivial topological manifold. The topological phases generalized on wavevectors are determined by the Gauss–Bonnet–Chern formula, in consideration of the mathematical structure of the 3D Ising model. Hence we prove the Zhang’s conjecture 2 (main theorem). Finally, we utilize the ferromagnetic 3D Ising model as a platform for describing a sensible interplay between the physical properties of many-body interacting systems, algebra, topology, and geometry.
format	article
author	Zhidong Zhang Osamu Suzuki
author_facet	Zhidong Zhang Osamu Suzuki
author_sort	Zhidong Zhang
title	A Method of the Riemann–Hilbert Problem for Zhang’s Conjecture 2 in a Ferromagnetic 3D Ising Model: Topological Phases
title_short	A Method of the Riemann–Hilbert Problem for Zhang’s Conjecture 2 in a Ferromagnetic 3D Ising Model: Topological Phases
title_full	A Method of the Riemann–Hilbert Problem for Zhang’s Conjecture 2 in a Ferromagnetic 3D Ising Model: Topological Phases
title_fullStr	A Method of the Riemann–Hilbert Problem for Zhang’s Conjecture 2 in a Ferromagnetic 3D Ising Model: Topological Phases
title_full_unstemmed	A Method of the Riemann–Hilbert Problem for Zhang’s Conjecture 2 in a Ferromagnetic 3D Ising Model: Topological Phases
title_sort	method of the riemann–hilbert problem for zhang’s conjecture 2 in a ferromagnetic 3d ising model: topological phases
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/15656cb6bb6441989ff0a35f6c3eb7fa
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A Method of the Riemann–Hilbert Problem for Zhang’s Conjecture 2 in a Ferromagnetic 3D Ising Model: Topological Phases

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