Beyond Integrability

There is one final problem of the hidden field of integrable models which I want to discuss. Namely the question of what is the relation of an integrable model to a generic physical system which does not satisfy a Yang-Baxter equation. For much of my career I have been told by many that these models are just mathematical curiosities which because they are integrable can, by that very fact have nothing to do with real physics. But on the other hand the fact remains that all of the phenomenological insight we have into real physics phase transitions as embodied in the notions of critical exponents, scaling theory and universality which have served us well for 35 years either all come from integrable models or are all confirmed by the the solutions of integrable models. So if integrable models leave something out we have a very poor idea of what it is.

Therefore it is greatly interesting that several months ago Bernie Nickel [41] sent around a preprint in which he made the most serious advance in the study of the Ising model susceptibility since our 1976 paper [14]. In that paper in addition to the Painlevé representation of the two point function we derive an infinite series for th Ising model susceptibility where the nth term in the series involves an nth order integral.

When the integrals in this expansion are scaled to the critical point each term contribute to the leading singularity of the susceptibility |T-T_c|^-7/4 However Nichols goes far beyond this scaling and for the isotropic case where E^v=E^h=E in term of the variable $v=\sinh 2E/kT$ he shows that successive terms in the series contribute singularities that eventually become dense on the unit circle in the complex plane |v|=1. From this he concludes that unless unexpected cancelations happen that there will be natural boundaries in the susceptibility on |v|=1. This would indeed be a new effect which could make integrable models different from generic models, Such natural boundaries have been suggested by several authors in the past including Guttmann [42], and Orrick and myself [43] on the basis of perturbation studies of nonintegrable models which show ever increasingly complicated singularity structures as the order of perturbation increases; a complexity which magically disappears when an integrability condition is imposed. This connection between integrability and analyticity was first emphasized by Baxter [44] long ago in 1980 when he emphasized that the Ising model in a magnetic field satisfies a functional equation very analogous to the zero field Ising model but that the Ising model in a field lacks the analyticity properties need for a solution. The rigorous proof of Nickel's conjecture will, if correct, open up a new view on what it means to be integrable.