Conjectures on exact solution of three - dimensional (3D) simple orthorhombic Ising lattices - Condensed Matter > Statistical MechanicsReport as inadecuate




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Abstract: We report the conjectures on the three-dimensional (3D) Ising model on simpleorthorhombic lattices, together with the details of calculations for a putativeexact solution. Two conjectures, an additional rotation in the fourth curled-updimension and the weight factors on the eigenvectors, are proposed to serve asa boundary condition to deal with the topologic problem of the 3D Ising model.The partition function of the 3D simple orthorhombic Ising model is evaluatedby spinor analysis, by employing these conjectures. Based on the validity ofthe conjectures, the critical temperature of the simple orthorhombic Isinglattices could be determined by the relation of KK* = KK- + KK- + K-K- orsinh 2K sinh 2(K- + K- + K-K-K) = 1. For a simple cubic Ising lattice, thecritical point is putatively determined to locate exactly at the golden ratioxc = exp(-2Kc) = (sq(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 orthorhombicIsing system would show a logarithmic singularity at the critical point of thephase transition. The spontaneous magnetization and the spin correlationfunctions of the simple orthorhombic Ising ferromagnet are derived explicitly.The putative critical exponents derived explicitly for the simple orthorhombicIsing lattices are alpha = 0, beta = 3-8, gamma = 5-4, delta = 13-3, eta = 1-8and nu = 2-3, showing the universality behavior and satisfying the scalinglaws. The cooperative phenomena near the critical point are studied and theresults obtained based on the conjectures are compared with those of theapproximation methods and the experimental findings. The 3D to 2D crossoverphenomenon differs with the 2D to 1D crossover phenomenon and there is agradual crossover of the exponents from the 3D values to the 2D ones.



Author: Zhi-dong Zhang

Source: https://arxiv.org/







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