Chiral Potts model

The Chiral Potts model is a spin model on a planar lattice in statistical mechanics. As with the Potts model, each spin can take n=0,...N-1 values. To each pair of nearest neighbor of spins n and n', a Boltzmann weight W(n-n') (Boltzmann factor) is assigned. The model is chiral, meaning W(n-n')≠ W(n'-n). When its weights satisfy the Yang-Baxter equation, (or the star-triangle relation), it is integrable. For the integrable Chiral Potts model, its weights are parametrized by a high genus curve, the Chiral Potts curve. Unlike the other solvable models, whose weights are parametrized by curves of genus less or equal to one, so that they can be expressed in term of trigonometric, or rational function (genus=0) or by theta functions (genus=1), this model involves high genus theta functions, which are not yet well developed. Therefore, it was thought that no progress could be made for such a difficult problem. Yet, many breakthroughs have been made since the 1990s. It must be stressed again that the Chiral Potts model was not invented because it was integrable but the integrable case was found, after it was introduced to explain experimental data. In a very profound way physics is here far ahead of mathematics. The history and its development will be presented here briefly.

Note that the chiral clock model, which has been introduced in the 80s of the last century, independently, by David Huse and Stellan Ostlund, is not exactly solvable, in contrast to the chiral Potts model.

This model is out of the class of all previously known models and raises a host of unsolved questions which are related to some of the most intractable problems of algebraic geometry which have been with us for 150 years. The chiral Potts models are used to understand the commensurate-incommensurate phase transitions.. For N = 3 and 4, the integrable case was discovered in 1986 in Stony Brook and published the following year.

The model is called self-dual, if the Fourier transform of the weight is equal to the weight. A special (genus 1) case had been solved in 1982 by Fateev and Zamolodchikov. By removing certain restrictions of the work of Alcaraz and Santos, a more general self-dual case of the integrable chiral Potts model was discovered. The weight are given in product form and the parameters in the weight are shown to be on the Fermat curve, with genus greater than 1.

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