“The statistical theory of phase changes in solids and liquids involves formidable mathematical problems.” Thus Lars Onsager began his 1944 magnum opus on the two-dimensional Ising model. As if to justify his opening line, he filled the 33 pages that followed with the exposition of a new algebra and the derivation of a crystal’s specific heat, partition function, and critical temperature. 1
The Ising model started life in 1925 as a simple one-dimensional quantum mechanical model of ferromagnetism. Each spin in a chain interacts with its two nearest neighbors and aligns with them, or not, depending on the interaction energy, the temperature, and chance.
Onsager generalized the Ising model to two dimensions, but his 1944 paper didn’t address what was in a sense the model’s original raison d’être: a solution for the spontaneous magnetization or its dimensionless equivalent, the order parameter M. He presented a solution without a derivation...