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

Abstract

This paper is a Response to Professor Perk's recent Comment (arXiv:1209.0731).We point out that the singularities of the reduced free energy βf , the free energy per site f and the free energy F of the 3D Ising model differ at β = 0. The rigorous proof presented in the Perk's Comment is only for the analyticity of the reduced free energy βf, which loses its definition at β = 0. Therefore, all of his objections lose the mathematical basis, which are thoroughly disproved. This means that the series expansions cannot serve as a standard for judging the correctness of the exact solution of the 3D Ising model. Furthermore, we note that there have been no comments on the topology-based approach developed by Zhang for the exact solution of the 3D Ising model.

Comments concerning the Ising model and two letters by N.H. March

“…After several theorists including the two of us had patiently explained errors in the drafts, Zhang shockingly got an enlarged version published [1]. That the result was in error was pointed out by Wu, McCoy, Fisher and Chayes [2], who showed that Zhang's formula for the free energy could not reproduce the known high-temperature series expansion while, at low temperatures, his results likewise failed, disagreeing with C.N.…”

“…Even though these values had been suggested in the long ago past [10,11], they were based on relatively short series expansions and are no longer supported by the best research. March thus had to appeal to the erroneous works of Zhang [1] to support these values.…”

The Magnetic Equation of State and Transport Properties in Reduced Dimensions

“…By analogy, we can write the expression for the spontaneous magnetization for a homogeneous 3D system [16] as…”

An Approach to Models of Order-Disorder and Ising Lattices

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The present paper develops the approach to the famous problem presented by L. Onsager (1944) and its further investigation proposed in a recent work by Z.-D. Zhang (2007). The above works give quaternion-based two-and three-dimensional (quantum) models of order-disorder transition and simple orthorhombic Ising lattices (1925).

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