Chiral Potts model

The chiral Potts model is a spin model on a planar lattice in statistical mechanics studied by Helen Au-Yang Perk and Jacques Perk, among others.

When the weights satisfy the Yang–Baxter equation, it is integrable, in the sense that certain quantities can be exactly evaluated.

The related chiral clock model, which was introduced in the 1980s by David Huse and Stellan Ostlund independently, is not exactly solvable, in contrast to the chiral Potts model.

The chiral Potts models are used to understand the commensurate-incommensurate phase transitions.

[5] For N = 3 and 4, the integrable case was discovered in 1986 in Stony Brook and published the following year.

A special (genus 1) case had been solved in 1982 by Fateev and Zamolodchikov.

[7] By removing certain restrictions of the work of Alcaraz and Santos,[8] a more general self-dual case of the integrable chiral Potts model was discovered.

[2] The weights were also given in product form and it was tested computationally (on Fortran) that they satisfy the star–triangle relation.

[11] From the series[5][12] the order parameter was conjectured[13] 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 proven by Baxter in 2005[14][15] using functional equations and the "broken rapidity line" technique of Jimbo et al.[16] assuming two mild analyticity conditions of the type commonly used in the field of Yang–Baxter integrable models.

In 1990 Bazhanov and Stroganov[24] showed that there exist L-operators (Lax operator) which satisfy the Yang–Baxter equation where the 2 × 2 R-operator (R-matrix) is the six vertex model R-matrix (see Vertex model).

The product of four chiral Potts weight S was shown to intertwine two L-operators as This inspired a breakthrough, namely the functional relations for the transfer matrices of the chiral Potts models were discovered.

[25] Using these functional relations, Baxter was able to calculate the eigenvalues of the transfer matrix of the chiral Potts model,[26] and obtained the critical exponent for the specific heat α=1-2/N, which was also conjectured in reference 12.

[27][28] The integrable chiral Potts weights are given in product form [2] as where

It was also known that the weights satisfy the inversion relation, This is equivalent to Reidemeister move II.

The star-triangle relation is equivalent to Reidemeister move III.