Exact Solution of Ferromagnetic Three-Dimensional (3D) Ising Model and Spontaneous Emerge of Time

doi:10.11900/0412.1961.2022.00486

Acta Metall Sin

2023, Vol. 59

Issue (4): 489-501 DOI: 10.11900/0412.1961.2022.00486

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Exact Solution of Ferromagnetic Three-Dimensional (3D) Ising Model and Spontaneous Emerge of Time

ZHANG Zhidong(

)

Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, China

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ZHANG Zhidong. Exact Solution of Ferromagnetic Three-Dimensional (3D) Ising Model and Spontaneous Emerge of Time. Acta Metall Sin, 2023, 59(4): 489-501.

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Abstract

This article reviews recent advances in the exact solution of ferromagnetic three-dimensional (3D) Ising model. First, the topological quantum statistic mechanism was introduced, which includes the time average, Jordan-von Neumann-Wigner framework, and the contribution of topological structures to thermodynamic properties of the system. Then, the Clifford algebra approach and the method of the Riemann-Hilbert problem were introduced to prove Zhang's two conjectures for the exact solution of the ferromagnetic 3D Ising model. The proof process verifies the correctness of the Zhang's exact solution for the ferromagnetic 3D Ising model. Based on these progresses, the origin of time was investigated and driven to the conclusion that in 3D many-body interacting particle (or spin) systems, time emerges spontaneously from many-body interactions.

Key words: Ising model exact solution topological quantum statistic time many-body interaction

Received: 08 October 2022

ZTFLH:

O414.21

Fund: National Natural Science Foundation of China(52031014)

Corresponding Authors: ZHANG Zhidong, professor, Tel: (024)23971859, E-mail: zdzhang@imr.ac.cn

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https://www.ams.org.cn/EN/10.11900/0412.1961.2022.00486 OR https://www.ams.org.cn/EN/Y2023/V59/I4/489

Fig.1 Schemes illustrate three of the braids in the transfer matrix V₃ connecting to the lattice points of the knot γ^[21] (The circles of V₁ and V₂ are not shown for simplicity. Also for simplicity, the ends of the three braids are not shown to connect with the nearest neighboring lattice points along the third dimension, P₁, P₂, …, P_n are lattice points)

Fig.2 Mapping between a state of crossings and a state of Ising spins^[22] (Two crossing states are mapped to two spin alignment states (up and down) with values of +1 and -1, respectively)

Fig.3 Mapping between the 2D-knot diagram with randomly distributed crossings on a lattice and a 2D Ising spin dual lattice^[22]

Fig.4 A braid is mapped to a spin-chain lattice^[22] (P and Q represent the ends of the braid or the spin chain, respectively)

Fig.5 Schemes illustrate a unit cell of the 3D cubic Ising lattice^[22] (All the five figures can be mapped each other, in which there are two contributions to physical properties: one is local spin alignment, another is nonlocal effect of the braids)
(a) with spins (red arrows) located at every lattice site and braids (blue curves) attached to every pair of the two nearest neighboring lattice sites along the third dimension
(b) with spins (red arrows) located at every lattice site and spin chains (black arrows with blue curves) formed by the mapping from the nontrivial knots (nonlinear terms)
(c) with crossings (red curves) are mapped from spins in a unit cell of the 3D Ising lattice, and some crossings (blue curves) are attached braids from the nontrivial topological structure
(d) a unit cell of the (3 + 1)D hypercubic Ising lattice in which the spins on the original 3D Ising lattice (black lines) are represented by red arrows and the spins in the high dimensional lattice (blue lines) are represented by black arrows
(e) a example of knots with crossings located on a unit cell of the (3 + 1)D hypercubic Ising lattice.

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