Mathematical structure of the three-dimensional (3D) Ising model

doi:10.1088/1674-1056/22/3/030513

摘要/Abstract

摘要： An overview of the mathematical structure of the three-dimensional (3D) Ising model is given from the viewpoints of topology, algebra, and geometry. By analyzing the relationships among transfer matrices of the 3D Ising model, Reidemeister moves in the knot theory, Yang-Baxter and tetrahedron equations, the following facts are illustrated for the 3D Ising model. 1) The complexified quaternion basis constructed for the 3D Ising model naturally represents the rotation in a (3+1)-dimensional space-time as a relativistic quantum statistical mechanics model, which is consistent with the 4-fold integrand of the partition function obtained by taking the time average. 2) A unitary transformation with a matrix that is a spin representation in 2^n·l·o-space corresponds to a rotation in 2n·l·o-space, which serves to smooth all the crossings in the transfer matrices and contributes the non-trivial topological part of the partition function of the 3D Ising model. 3) A tetrahedron relation would ensure the commutativity of the transfer matrices and the integrability of the 3D Ising model, and its existence is guaranteed by the Jordan algebra and the Jordan-von Neumann-Wigner procedures. 4) The unitary transformation for smoothing the crossings in the transfer matrices changes the wave functions by complex phases φ_x, φ_y, and φ_z. The relationship with quantum field and gauge theories and the physical significance of the weight factors are discussed in detail. The conjectured exact solution is compared with numerical results, and the singularities at/near infinite temperature are inspected. The analyticity in β=1/(k_BT) of both the hard-core and the Ising models has been proved only for β>0, not for β=0. Thus the high-temperature series cannot serve as a standard for judging a putative exact solution of the 3D Ising model.

关键词: Ising model, topology, algebra, geometry

Abstract: An overview of the mathematical structure of the three-dimensional (3D) Ising model is given from the viewpoints of topology, algebra, and geometry. By analyzing the relationships among transfer matrices of the 3D Ising model, Reidemeister moves in the knot theory, Yang-Baxter and tetrahedron equations, the following facts are illustrated for the 3D Ising model. 1) The complexified quaternion basis constructed for the 3D Ising model naturally represents the rotation in a (3+1)-dimensional space-time as a relativistic quantum statistical mechanics model, which is consistent with the 4-fold integrand of the partition function obtained by taking the time average. 2) A unitary transformation with a matrix that is a spin representation in 2^n·l·o-space corresponds to a rotation in 2n·l·o-space, which serves to smooth all the crossings in the transfer matrices and contributes the non-trivial topological part of the partition function of the 3D Ising model. 3) A tetrahedron relation would ensure the commutativity of the transfer matrices and the integrability of the 3D Ising model, and its existence is guaranteed by the Jordan algebra and the Jordan-von Neumann-Wigner procedures. 4) The unitary transformation for smoothing the crossings in the transfer matrices changes the wave functions by complex phases φ_x, φ_y, and φ_z. The relationship with quantum field and gauge theories and the physical significance of the weight factors are discussed in detail. The conjectured exact solution is compared with numerical results, and the singularities at/near infinite temperature are inspected. The analyticity in β=1/(k_BT) of both the hard-core and the Ising models has been proved only for β>0, not for β=0. Thus the high-temperature series cannot serve as a standard for judging a putative exact solution of the 3D Ising model.

Key words: Ising model, topology, algebra, geometry

中图分类号: (Lattice theory and statistics)

05.50.+q

75.10.Hk (Classical spin models) 05.70.Fh (Phase transitions: general studies)

张志东. Mathematical structure of the three-dimensional (3D) Ising model[J]. 中国物理B, 2013, 22(3): 30513-030513.

Zhang Zhi-Dong (张志东). Mathematical structure of the three-dimensional (3D) Ising model[J]. Chin. Phys. B, 2013, 22(3): 30513-030513.

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