Conjectures on the exact solution of three-dimensional (3D) simple orthorhombic Ising lattices

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	Conjectures on the exact solution of three-dimensional (3D) simple orthorhombic Ising lattices
	Z. D. Zhang
	2007
发表期刊	Philosophical Magazine
ISSN	1478-6435
卷号	87 期号:34 页码:5309-5419
摘要	We report conjectures on the three-dimensional (3D) Ising model of simple orthorhombic lattices, together with details of calculations for a putative exact solution. Two conjectures, an additional rotation in the fourth curled-up dimension and weight factors on the eigenvectors, are proposed to serve as a boundary condition to deal with the topologic problem of the 3D Ising model. The partition function of the 3D simple orthorhombic Ising model is evaluated by spinor analysis, employing these conjectures. Based on the validity of the conjectures, the critical temperature of the simple orthorhombic Ising lattices could be determined by the relation of KK* = KK ' + KK '' + K ' K '' or sinh 2K - sinh 2(K ' + K '' + (K ' K ''/K)) = 1. For a simple cubic Ising lattice, the critical point is putatively determined to locate exactly at the golden ratio x(c) = e(-2kc) = ((root 5 - 1)/2), as derived from K* = 3K or sinh 2K = sinh 6K = 1. If the conjectures would be true, the specific heat of the simple orthorhombic Ising system would show a logarithmic singularity at the critical point of the phase transition. The spontaneous magnetization of the simple orthorhombic Ising ferromagnet is derived explicitly by the perturbation procedure, following the conjectures. The spin correlation functions are discussed on the terms of the Pfaffians, by defining the effective skew-symmetric matrix A(eff). The true range k(x) of the correlation and the susceptibility of the simple orthorhombic Ising system are determined by procedures similar to those used for the two-dimensional Ising system. The putative critical exponents derived explicitly for the simple orthorhombic Ising lattices are alpha = 0, beta = 3/8, gamma = 5/4, delta = 13/3, eta = 1/8 and nu = 2/3, showing the universality behaviour and satisfying the scaling laws. The cooperative phenomena near the critical point are studied and the results based on the conjectures are compared with those of approximation methods and experimental findings. The putative solutions have been judged by several criteria. The deviations of the approximation results and the experimental data from the solutions are interpreted. Based on the solution, it is found that the 3D-to-2D crossover phenomenon differs with the 2D-to-1D crossover phenomenon and there is a gradual crossover of the exponents from the 3D to the 2D values. Special attention is also paid to the extra energy caused by the introduction of the fourth curled-up dimension and the states at/near infinite temperature revealed by the weight factors of the eigenvectors. The physics beyond the conjectures and the existence of the extra dimension are discussed. The present work is not only significant for statistical and condensed matter physics, but also fill the gap between the quantum field theory, cosmology theory, high-energy particle physics, graph theory and computer science.
部门归属	chinese acad sci, met res inst, shenyang natl lab mat sci, shenyang 110016, peoples r china. chinese acad sci, int ctr mat phys, shenyang 110016, peoples r china.;zhang, zd (reprint author), chinese acad sci, met res inst, shenyang natl lab mat sci, 72 wenhua rd, shenyang 110016, peoples r china;zdzhang@imr.ac.cn
关键词	Monte-carlo Renormalization Critical-point Scattering Order-disorder Problems Low-temperature Series Super-exchange Antiferromagnet Dynamic Critical Phenomena Quantum Phase-transitions Centered-cubic Lattices Feynman-graph Expansion Excluded-volume Problem
URL	查看原文
WOS记录号	WOS:000250777300001
引用统计	被引频次：110[WOS] [WOS记录] [WOS相关记录]
文献类型	期刊论文
条目标识符	http://ir.imr.ac.cn/handle/321006/34061
专题	中国科学院金属研究所
推荐引用方式 GB/T 7714	Z. D. Zhang. Conjectures on the exact solution of three-dimensional (3D) simple orthorhombic Ising lattices[J]. Philosophical Magazine,2007,87(34):5309-5419.
APA	Z. D. Zhang.(2007).Conjectures on the exact solution of three-dimensional (3D) simple orthorhombic Ising lattices.Philosophical Magazine,87(34),5309-5419.
MLA	Z. D. Zhang."Conjectures on the exact solution of three-dimensional (3D) simple orthorhombic Ising lattices".Philosophical Magazine 87.34(2007):5309-5419.