Next: The chiral Potts model Up: No Title Previous: Ising model in a

From Ising to integrable

At the time when most of this Ising model work was being done there were only a few other exactly solved models known: The delta function gases, the six vertex model, the Hubbard model, and the 8 vertex model. Solution for these models meant the computation of the free energy ( or ground state energy), the excitation spectrum and the order parameter. But starting in the late 70's it was realized that a fundamental equation first seen by Onsager [1]-[23] in the Ising model and used in a profound way by Yang [24] in the delta function gases and by Baxter [25] in the 8 vertex model could be used to find many large classes of models for which free energies could be computed. These models which come from the Yang-Baxter (or star triangle equation) are what are now called the integrable models.

The Ising model itself is the simplest case of such an integrable model. It thus seems to be a very natural conjecture which was made by Wu, myself and our collaborators the instant we made the discovery of the Painlevé representation of the Ising correlation function that there must be a similar representation for the correlation functions of all integrable models. To be more precise I mean the following

Conjecture

The correlation functions of integrable statistical mechanical models are characterized as the solutions of classically integrable equations (be they differential, integral or difference).

One major step in the advancement of this program was made by our next speaker, Sasha Zamolodchikov, who showed, with the invention of conformal field theory [26], that this conjecture is realized for models at the critical point. One of the major unsolved problems of integrable models today is to extend the linear equations which characterize correlation functions in conformal field theory to nonlinear equations for massive models. This will realize the goal of generalizing to all integrable models what we have learned for the correlation functions of the Ising model. This is an immense field of undertaking in which many people have and are making major contributions. It is surely not possible to come close to surveying this work in the few minutes left to me. I will therefore confine myself to a few remarks about things I have been personally involved with since completing the work with Wu in 1981 on the Ising model.

Next: The chiral Potts model Up: No Title Previous: Ising model in a
Barry McCoy
3/29/1999