Response to arXiv:1209.0731 'Erroneous solution of three-dimensional (3D) simple orthorhombic Ising lattices' by Perk

TL;DR

This paper refutes a recent critique of the 3D Ising model's solution, clarifying the analyticity issues at zero temperature and emphasizing the importance of topology-based approaches for understanding the model.

Contribution

It provides a rigorous rebuttal to Perk's objections, clarifies the analyticity properties of the free energy at zero temperature, and highlights the significance of topology-based methods in solving the 3D Ising model.

Findings

Perk's proof only applies for β > 0, not at β = 0

Series expansions are not definitive for exact solutions

Topology-based approach offers a promising alternative

Abstract

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 the analyticity in {\beta} of both the hard-core and Ising models is proved for {\beta} > 0, not for {\beta} = 0. 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.