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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.〔Au-Yang H, McCoy B M, Perk J H H, Tang S and Yan M-L (1987), "Commuting transfer matrices in the chiral Potts models: Solutions of the star-triangle equations with genus > 1", Physics Letters A 123 219–23.〕〔Baxter R J, Perk J H H and Au-Yang H (1988), "New solutions of the star-triangle relations for the chiral Potts model", Physics Letters A 128 138–42.〕 Unlike the other solvable models,〔R. J. Baxter,"Exactly Solved Models in Statistical Mechanics", Academic Press, ISBN 978-0-12-083180-7.〕〔B.M.McCoy,"Advanced Statistical Mechanics", 146 International Series of Monographs on Physics, Oxford, England, ISBN 9780199556632〕 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. == 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.〔S. Howes, L.P. Kadanoff and M. den Nijs (1983), Nuclear Physics B 215, 169.〕〔Julia M. Yeomans, Michael E. Fisher (1984), "Analysis of the multiphase region in the three-state chiral clock model", Physica A 127, 1–37.〕 For N = 3 and 4, the integrable case was discovered in 1986 in Stony Brook and published the following year.〔〔McCoy B M, Perk J H H, Tang S and Sah C H (1987), "Commuting transfer matrices for the 4 state self-dual chiral Potts model with a genus 3 uniformizing Fermat curve", Physics Letters A 125, 9–14.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Chiral Potts model」の詳細全文を読む スポンサード リンク
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