It was a great pleasure to hear that the American Physical Society had awarded me the 2006 Lars Onsager Prize and to read the informative news story “Order Parameter of the Chiral Potts Model Succumbs at Last to Exact Solution” that appeared on page 19 of the November edition of Physics Today. I have a few comments.
There are various models named after Ren Potts, who died in Adelaide, Australia, on 9 August 2005. He was a student of Cyril Domb at Oxford University in 1952, and worked on the vector and scalar extensions of the Ising model. 1 The two are often confused. The vector model is described in the Physics Today story, whereas it is the scalar model that has been solved at criticality. The N-state chiral Potts model is a different model again. Its transfer matrices commute, which makes it “solvable” in the sense that one can calculate the free energy. For N = 2 it reduces to the Ising model.
It is true that C. N. Yang took six months to calculate the spontaneous magnetization (order parameter) of the Ising model, 2 whereas I have been puzzling over the extension of that problem to the chiral Potts model for 15 years!
For both models the results were previously known: Onsager had announced his result for the Ising model at a conference in 1949, 3 and Giuseppe Albertini, Barry McCoy, Jacques Perk, and Shuang Tang gave the formula for chiral Potts in 1989. However, there are some differences.
The story refers to both these announcements as conjectures, but it is clear from a later paper by Onsager 4 that he had already derived his answer when he announced it. Being Onsager, he did not rush into print, but worked at refining and polishing his derivation for three years, which gave Yang the opportunity to publish his solution in 1952.
By contrast, the chiral Potts result given by Giuseppe Albertini and colleagues really was a conjecture—an elegant formula, inspired by the Ising result, that fitted all known series expansions. It is this conjecture that I have now verified, subject to analyticity assumptions no more drastic than those used for the order parameters of other post-Ising models.
From 1998 until recently, I was looking at the Jimbo-Miwa-Nakayashiki method. It depends on solving for a function of two arguments, and I was having trouble. The idea that came to me over two days in December 2004 was to relate those arguments so that there was just one independent variable, in much the same way as in the free energy calculation one starts by looking at the simpler “superintegrable” case. The switch to one variable made the needed analyticity properties fairly obvious, and was sufficient to obtain the order parameters. 5 It was a good illustration of the approach used in these solvable models—one needs to generalize sufficiently to see properties like commuting transfer matrices, but one should beware of generalizing unnecessarily.
