The chiralPotts 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 is assigned. The model is chiral, meaning W≠ W. When its weights satisfy the Yang–Baxter equation,, 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 or by theta functions, 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 1980s, independently, by David Huse and Stellan Ostlund, is not exactly solvable, in contrast to the chiral Potts model.
The 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 Stonybrook and published the following year.
Self-dual case
The model is called self-dual, if the Fourier transform of the weight is equal to the weight. A special 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.
General case
In Canberra, the general solution for all k was found. The weights were also given in product form and it was tested by Fortran that they satisfy the star–triangle relation. The proof was published later.
Results
Order parameter
From the series the order parameter is conjectured to have the simple form It took many years to prove this conjecture, as the usual corner transfer matrix technique could not be used, because of the higher genus curve. This conjecture was finally proven by Baxter in 2005 using functional equations and the "broken rapidity line" technique of Jimbo et al. assuming two rather mild analyticity conditions of the type commonly used in the field of Yang–Baxter integrable models. Most recently, in a series of papers an algebraic way of obtaining order parameter has been given, giving more insight into the algebraic structure.
In 1990 Bazhanov and Stroganov show that there exist an 2 × N L-operators, which satisfy the Yang–Baxter equation where 2 × 2 R-operator is the 6-vertex R-matrix. The product of four chiral Potts weight S was shown to intertwine two L-operators as This inspired a most important breakthrough, namely the functional relations for the transfer matrices of the chiral Potts models are discovered.
The integrable chiral Potts weights are given in product form as where ωN=1 and we associate with each of the rapidity variable p with three variables satisfying It is easy to see that which is similar to Reidemeister move I. It was also known that the weights satisfying the inversion relation, This is equivalent to Reidemeister move II. The star-triangle relation is equivalent to Reidemeister move III. These are shown in the figure shown here.
