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

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.

Topological Quantum Statistical Mechanics and Topological Quantum Field Theories

“…Any procedures that do not consider such contributions cannot derive a correct solution, owing to lack of important energy terms arising from topology. This is also why approximations (such as conventional low-temperature expansions, conventional high-temperature expansions, renormalization group, and Monte Carlo simulations) without consideration of nontrivial topological contributions cannot serve as a standard for adjudging the correctness of an exact solution [10][11][12][13].…”

“…This kind of knot, which can be either nontrivial or trivial, represents the local spin alignments at the lattice points and the nearest neighboring interactions between spins (edges connecting points). The linear terms of Γ matrices in the transfer matrices V1 and V2 correspond to circles (or intervals), while the internal factors of nonlinear terms of Γ matrices in V3 correspond to braids [12]. The circles (or intervals) and braids are attached on every lattice point.…”

“…Zhang, Suzuki, and March proved four theorems (Trace Invariance Theorem, Linearization Theorem, Local Transformation Theorem, Commutation Theorem) in [11], which rigorously proved Zhang's two conjectures. Furthermore, Suzuki and Zhang [12,13] rigorously proved the two conjectures by the method of the Riemann-Hilbert problem, the monoidal transformation, and the Gauss-Bonnet-Chern formula. Recently, the lower bound of the computational complexity of the spin-glass 3D Ising model was determined [14].…”

“…Analogously, in some models of quantum field theory (QFT), nontrivial topological structures exist also, which can be defined as topological quantum field theory (TQFT). Although the 3D Ising model is one of the simplest many-body interacting models, the findings in our previous work [10][11][12][13] may be applicable for other many-body interacting systems. It will be extremely important to catalog the models in the same class and/or with the same mathematical structure.…”

“…The exact solution of the ferromagnetic 3D Ising model stands up as a hard problem in physics for almost 100 years [8], since it is extremely difficult to overcome several key obstacles, such as nonlocal behavior (nontrivial topological structures), non-Gaussian behavior (nonlinear terms of Γ-matrices), noncommutative behavior of Γ-operators [11]. In the procedures developed for solving explicitly the solution [10][11][12][13], we performed a gauge transformation or a monoidal transformation in a higher dimensional space-time to deal with the nonlocal problem, processed a linearization procedure to remove the non-Gaussian obstruction, and introduced the Jordan-von Neumann-Wigner (JNW) framework to eliminate the noncommutative hindrance. Because the 3D Ising model is a paradigm in TQSM, other models in TQSM may also possess these basic characters so that the same (nonlocal, non-Gaussian, and noncommutative) obstacles do exist for exactly solving them.…”