Zhidong Zhang, Norman H. March
This paper is a Response to Professor J.H.H. Perk's recent Comment (arXiv:1209.0731v1). We point out that the singularities of the reduced free energy {\beta}f, the free energy per site f and the free energy F of the 3D Ising model differ at {\beta} = 0. The rigorous proof presented in the Perk's Comment is only for the analyticity of the reduced free energy {\beta}f, which loses its definition at {\beta} = 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. A Rejoinder to Professor J.H.H. Perk's open letter in arXiv:1209.0731v2 is added. We show that singularities at {\beta} = 0 are different for the hard core model and the Ising model and that there is no upper bound at {\beta} = 0 for series of the Ising model. Furthermore, the free energy per site f and the reduced free energy {\beta}f lose their definitions at {\beta} = 0, and thus either of them could have two different forms for high-temperature series expansions at/below infinite temperature. Three independent Virasoro algebras for 3D conformal field theory can be written within the 3+1 dimensional space (i.e., 3-sphere) with weight factors.
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http://arxiv.org/abs/1209.3247
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