A master solution of the quantum Yang-Baxter equation and classical discrete integrable equations

摘要

We obtain a new solution of the star-triangle relation with positiveBoltzmann weights which contains as special cases all continuous and discretespin solutions of this relation, that were previously known. This new mastersolution defines an exactly solvable 2D lattice model of statistical mechanics,which involves continuous spin variables, living on a circle, and contains twotemperature-like parameters. If one of the these parameters approaches a rootof unity (corresponds to zero temperature), the spin variables freezes intodiscrete positions, equidistantly spaced on the circle. An absolute orientationof these positions on the circle slowly changes between lattice sites byoverall rotations. Allowed configurations of these rotations are described byclassical discrete integrable equations, closely related to the famous$Q_4$-equations by Adler Bobenko and Suris. Fluctuations between degenerateground states in the vicinity of zero temperature are described by a rathergeneral integrable lattice model with discrete spin variables. In some simplespecial cases the latter reduces to the Kashiwara-Miwa and chiral Potts models.