2021
DOI: 10.3390/math9070776
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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 r… Show more

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Cited by 5 publications
(27 citation statements)
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References 80 publications
(93 reference statements)
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“…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].…”
Section: Exact Solution Of the Ferromagnetic 3d Ising Modelmentioning
confidence: 99%
See 4 more Smart Citations
“…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].…”
Section: Exact Solution Of the Ferromagnetic 3d Ising Modelmentioning
confidence: 99%
“…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.…”
Section: Exact Solution Of the Ferromagnetic 3d Ising Modelmentioning
confidence: 99%
See 3 more Smart Citations