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 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.

Automated summary

The study employed the Riemann-Hilbert problem to address Zhang's conjecture 2 regarding the ferromagnetic Ising model in zero external magnetic field, proving that the 3D Ising model can be mapped to higher dimensions or trivial topological structures. It introduced vertex operators of knot types and a flat vector bundle for the model, and utilized monoidal transforms to trivialize knots/links in a 4D Riemann manifold. By applying the Gauss-Bonnet-Chern formula, the study demonstrated the relationship between topological phases and the mathematical structure of the 3D Ising model, ultimately proving Zhang's conjecture 2.