Abstract
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.
Highlights
The Ising model has attracted intensive interest since the 1920s [1], which applies to the interpretation of phase transitions and critical phenomena in different fields, and provides a fundamental understanding on interactions and dimensionality
Zhang conjectured that the nontrivial knot/link structures of the ferromagnetic 3D Ising model in the zero external magnetic field can be trivialized in higher dimensional space and it can be realized as the free statistic model on the (3 + 1)-dimensional (i.e., (3 + 1)D) space with topological/geometrical phases on eigenvectors [3,4]
In the preceding paper [10] of the present series, we developed a method of the Riemann–Hilbert problem for Zhang’s conjecture 1 proposed in [3] and rigorously proved Zhang’s conjecture 1 in the following steps [10]: (1) the Clifford algebra Cl(I3D) is extended to the Knot/Clifford (K/C) algebra which has the original Clifford algebra and its conjugate algebra Cl(I3D) as subalgebras [10]; and (2) The K/C knot Zγ is extended to the K/C algebra, which is denoted by σ(Zγ, Zγ)
Summary
The Ising model has attracted intensive interest since the 1920s [1], which applies to the interpretation of phase transitions and critical phenomena in different fields, and provides a fundamental understanding on interactions and dimensionality. The mapping provides a concise proof of Zhang’s conjectures 1 and 2 This procedure is helpful for understanding the contribution of nontrivial topological structures to the partition function and physical properties of the ferromagnetic 3D Ising model, and the spontaneous emergence of the additional dimension, and the connection between different approaches developed based on topology and algebra. As long as nontrivial knots or links exist in a system, as revealed in [4], a matrix representing such a transformation (i.e., a rotation) intrinsically and spontaneously may always exist, no matter how complicated the knots or links are This topology fact is employed in the recent work [7] to prove the Local Transformation Theorem, in order to trivialize the nontrivial topological structure in the 3D Ising model. One may refer to [3,7] for the results of these physical properties obtained by the local transformation
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