A master solution of the quantum Yang-Baxter equation and classical discrete integrable equations
Australian National University, Canberra
Fri, Oct. 07th 2011, 14:15
Salle Claude Itzykson, Bât. 774, Orme des Merisiers
We describe a new solution of the star-triangle relation with positive Boltzmann weights which contains as special cases all continuous and discrete spin solutions of this relation, that were previously known. This new master solution defines an exactly solvable 2D lattice model of statistical mechanics, which involves continuous spin variables, living on a circle, and contains two temperature-like parameters. If one of the these parameters approaches a root of unity (corresponds to zero temperature), the spin variables freezes into discrete positions, equidistantly spaced on the circle. An absolute orientation of these positions on the circle slowly changes between lattice sites by overall rotations. Allowed configurations of these rotations are described by classical discrete integrable equations, closely related to the famous $Q_4$-equations by Adler Bobenko and Suris. Fluctuations between degenerate ground states in the vicinity of zero temperature are described by a new ``hybrid'' model which combines a classical integrable system with continuous variables and a lattice model of statistical mechanics with discrete spins, where the Boltzmann weights depend on the classical solution. In some simple special cases the model reduces to the Kashiwara-Miwa and chiral Potts models.