Why are there so many solutions to the two-dimensional Ising model?

Abstract

Exact solutions for the partition function of the two-dimensional classical statistical mechanics Ising model may be classified using a scheme of analogy having three moments--analysis, algebra, and arithmetic--developed by Dedekind and Weber in 1882 for providing an algebraic understanding of Riemann’s work. In effect, we have two analogies, a physical one and a mathematical one, coming from very different problems. What is it about the mathematical realm that allows for this threefold analogy, and what is it about the Ising model that allows for such varied modes of solution, and why are the analogies analogous to each other?

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