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arXiv:cond-mat/0510683 · Journal of Physics Conference Series

The challenge of the chiral Potts model

R. J. Baxter

Published Jun 1, 2006• 9 citations
arXivHTMLPDF

Abstract

The chiral Potts model continues to pose particular challenges in statistical mechanics: it is ''exactly solvable'' in the sense that it satisfies the Yang-Baxter relation, but actually obtaining the solution is not easy. Its free energy was calculated in 1988 and the order parameter was conjectured in full generality a year later. However, a derivation of that conjecture had to wait until 2005. Here we discuss that derivation.

DOI
10.1088/1742-6596/42/1/003

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The challenge of the chiral Potts model

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The chiral Potts model continues to pose particular challenges in statistical mechanics: it is “exactly solvable ” in the sense that it satisfies the Yang-Baxter relation, but actually obtaining the solution is not easy. Its free energy was calculated in 1988 and the order parameter was conjectured in full generality a year later. However, a derivation of that conjecture had to wait until 2005. Here we discuss that derivation. 1

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It was a great pleasure to hear that the American Physical Society had awarded me the 2006 Lars Onsager Prize and to read the informative news story “Order Parameter of the Chiral Potts Model Succumbs at Last to Exact Solution” that appeared on page 19 of the November edition of Physics Today. I have a few comments.There are various models named after Ren Potts, who died in Adelaide, Australia, on 9 August 2005. He was a student of Cyril Domb at Oxford University in 1952, and worked on the vector and scalar extensions of the Ising model. 1 1. R. Potts, Proc. Camb. Phil. Soc. 48, 106 (1952) https://doi.org/10.1017/S0305004100027419. The two are often confused. The vector model is described in the Physics Today story, whereas it is the scalar model that has been solved at criticality. The N-state chiral Potts model is a different model again. Its transfer matrices commute, which makes it “solvable” in the sense that one can calculate the free energy. For N = 2 it reduces to the Ising model. It is true that C. N. Yang took six months to calculate the spontaneous magnetization (order parameter) of the Ising model, 2 2. C. N. Yang, Collected Papers, Freeman and Co, San Francisco (1983), p. 12. whereas I have been puzzling over the extension of that problem to the chiral Potts model for 15 years!For both models the results were previously known: Onsager had announced his result for the Ising model at a conference in 1949, 3 3. L. Onsager, Nuovo Cimento 6(suppl.), 261 (1949). and Giuseppe Albertini, Barry McCoy, Jacques Perk, and Shuang Tang gave the formula for chiral Potts in 1989. However, there are some differences.The story refers to both these announcements as conjectures, but it is clear from a later paper by Onsager 4 4. L. Onsager, in Critical Phenomena in Alloys, Magnets and Superconductors, R. E. Mills, E. Ascher, R. I. Jaffee, eds., McGraw-Hill, New York (1971), p. 3. that he had already derived his answer when he announced it. Being Onsager, he did not rush into print, but worked at refining and polishing his derivation for three years, which gave Yang the opportunity to publish his solution in 1952.By contrast, the chiral Potts result given by Giuseppe Albertini and colleagues really was a conjecture—an elegant formula, inspired by the Ising result, that fitted all known series expansions. It is this conjecture that I have now verified, subject to analyticity assumptions no more drastic than those used for the order parameters of other post-Ising models. From 1998 until recently, I was looking at the Jimbo-Miwa-Nakayashiki method. It depends on solving for a function of two arguments, and I was having trouble. The idea that came to me over two days in December 2004 was to relate those arguments so that there was just one independent variable, in much the same way as in the free energy calculation one starts by looking at the simpler “superintegrable” case. The switch to one variable made the needed analyticity properties fairly obvious, and was sufficient to obtain the order parameters. 5 5. R. J. Baxter, J. Stat. Phys. 120, 1 (2005) https://doi.org/10.1007/s10955-005-5534-3. It was a good illustration of the approach used in these solvable models—one needs to generalize sufficiently to see properties like commuting transfer matrices, but one should beware of generalizing unnecessarily.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. R. Potts, Proc. Camb. Phil. Soc. 48, 106 (1952) https://doi.org/10.1017/S0305004100027419. Google ScholarCrossref, ISI2. C. N. Yang, Collected Papers, Freeman and Co, San Francisco (1983), p. 12. Google Scholar3. L. Onsager, Nuovo Cimento 6(suppl.), 261 (1949). Google ScholarCrossref4. L. Onsager, in Critical Phenomena in Alloys, Magnets and Superconductors, R. E. Mills, E. Ascher, R. I. Jaffee, eds., McGraw-Hill, New York (1971), p. 3. Google Scholar5. R. J. Baxter, J. Stat. Phys. 120, 1 (2005) https://doi.org/10.1007/s10955-005-5534-3. Google ScholarCrossref, ISI© 2005 American Institute of Physics.

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Proceedings of the conference on exactly soluble models in statistical mechanics : historical perspectives and current status, Northeastern University, Boston 30-31 March 1996

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The chiral Potts model continues to pose particular challenges in statistical mechanics: it is ''exactly solvable'' in the sense that it satisfies the Yang-Baxter relation, but actually obtaining the solution is not easy. Its free energy was calculated in 1988 and the order parameter was conjectured in full generality a year later. However, a derivation of that conjecture had to wait until 2005. Here we discuss that derivation.
It was a great pleasure to hear that the American Physical Society had awarded me the 2006 Lars Onsager Prize and to read the informative news story “Order Parameter of the Chiral Potts Model Succumbs at Last to Exact Solution” that appeared on page 19 of the November edition of Physics Today. I have a few comments.There are various models named after Ren Potts, who died in Adelaide, Australia, on 9 August 2005. He was a student of Cyril Domb at Oxford University in 1952, and worked on the vector and scalar extensions of the Ising model. 1 1. R. Potts, Proc. Camb. Phil. Soc. 48, 106 (1952) https://doi.org/10.1017/S0305004100027419. The two are often confused. The vector model is described in the Physics Today story, whereas it is the scalar model that has been solved at criticality. The N-state chiral Potts model is a different model again. Its transfer matrices commute, which makes it “solvable” in the sense that one can calculate the free energy. For N = 2 it reduces to the Ising model. It is true that C. N. Yang took six months to calculate the spontaneous magnetization (order parameter) of the Ising model, 2 2. C. N. Yang, Collected Papers, Freeman and Co, San Francisco (1983), p. 12. whereas I have been puzzling over the extension of that problem to the chiral Potts model for 15 years!For both models the results were previously known: Onsager had announced his result for the Ising model at a conference in 1949, 3 3. L. Onsager, Nuovo Cimento 6(suppl.), 261 (1949). and Giuseppe Albertini, Barry McCoy, Jacques Perk, and Shuang Tang gave the formula for chiral Potts in 1989. However, there are some differences.The story refers to both these announcements as conjectures, but it is clear from a later paper by Onsager 4 4. L. Onsager, in Critical Phenomena in Alloys, Magnets and Superconductors, R. E. Mills, E. Ascher, R. I. Jaffee, eds., McGraw-Hill, New York (1971), p. 3. that he had already derived his answer when he announced it. Being Onsager, he did not rush into print, but worked at refining and polishing his derivation for three years, which gave Yang the opportunity to publish his solution in 1952.By contrast, the chiral Potts result given by Giuseppe Albertini and colleagues really was a conjecture—an elegant formula, inspired by the Ising result, that fitted all known series expansions. It is this conjecture that I have now verified, subject to analyticity assumptions no more drastic than those used for the order parameters of other post-Ising models. From 1998 until recently, I was looking at the Jimbo-Miwa-Nakayashiki method. It depends on solving for a function of two arguments, and I was having trouble. The idea that came to me over two days in December 2004 was to relate those arguments so that there was just one independent variable, in much the same way as in the free energy calculation one starts by looking at the simpler “superintegrable” case. The switch to one variable made the needed analyticity properties fairly obvious, and was sufficient to obtain the order parameters. 5 5. R. J. Baxter, J. Stat. Phys. 120, 1 (2005) https://doi.org/10.1007/s10955-005-5534-3. It was a good illustration of the approach used in these solvable models—one needs to generalize sufficiently to see properties like commuting transfer matrices, but one should beware of generalizing unnecessarily.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. R. Potts, Proc. Camb. Phil. Soc. 48, 106 (1952) https://doi.org/10.1017/S0305004100027419. Google ScholarCrossref, ISI2. C. N. Yang, Collected Papers, Freeman and Co, San Francisco (1983), p. 12. Google Scholar3. L. Onsager, Nuovo Cimento 6(suppl.), 261 (1949). Google ScholarCrossref4. L. Onsager, in Critical Phenomena in Alloys, Magnets and Superconductors, R. E. Mills, E. Ascher, R. I. Jaffee, eds., McGraw-Hill, New York (1971), p. 3. Google Scholar5. R. J. Baxter, J. Stat. Phys. 120, 1 (2005) https://doi.org/10.1007/s10955-005-5534-3. Google ScholarCrossref, ISI© 2005 American Institute of Physics.
Some of the early history of exactly soluble models, E.H. Lieb the many faces of the chiral Potts model, H. Au-Yang and J.H.H. Perk star-triangle and star-star relations in statistical mechanics, R.J. Baxter knot theory and statistical mechanics, L.H. Kauffman star-star relation for the Potts model, C. King and F.Y. Wu finite temperature correlations of the Ising chain in a transverse field, S. Sachdev exact solutions of low-dimensional reaction-diffusion systems, A.M.R. Cadilhe et al the infinite-state Potts model and solid partitions of an integer, H.Y. Huang and F.Y. Wu exact evaluation of thermodynamic quantities for the integrable T-J chain, A. Klumper multilayer extension of two-dimensional solvable statistical models to three dimensions, V. Popkov a higher-level Bailey lemma, A. Schilling and S.O. Warnaar the Heisenberg chain as a dynamical model for protein synthesis - some theoretical and experimental results, G.M. Schutz free fermion conditions and the symmetries of integrability, C.-M. Viallet. (Part contents).
We derive the free energy of the chiral Potts model by the infinite lattice “inversion relation” method. This method is non-rigorous in that it always needs appropriate analyticity assumptions. Guided by previous calculations based on exact finite-lattice functional relations, we find that in addition to the usual assumption that the free energy be analytic and bounded in some principal domain of the rapidity parameter space that includes the physical regime, we also need a much less obvious symmetry. We can then obtain the free energy by Wiener-Hopf ~factorization in the complex planes of appropriate variables. Together with the inversion relation, this symmetry relates the values of the free energy in all neighbouring domains to those in the principal domain.
In honor of Onsager’s ninetieth birthday, we like to review some exact results obtained so far in the chiral Potts models and to translate these results into language more transparent to physicists, so that experts in Monte Carlo calculations, high and low temperature expansions, and various other methods, can use them. We shall pay special attention to the interfacial tension ǫr between the k state and the k −r state. By examining the ground states, it is seen that the integrable line ends at a superwetting point, on which the relation ǫr = r ǫ1 is satisfied, so that it is energetically neutral to have one interface or more. We present also some partial results on the meaning of the integrable line for low temperatures where it lives in the non-wet regime. We make Baxter’s exact results more explicit for the symmetric case. By performing a Bethe Ansatz calculation with open boundary conditions we confirm a dilogarithm identity for the low-temperature expansion which may be new. We propose a new model for numerical studies. This model has only two variables and exhibits commensurate and incommensurate phase transitions and wetting transitions near zero temperature. It appears to be not integrable, except at one point, and at each temperature there is a point, where it is almost identical with the integrable chiral Potts model. KEY WORDS: Chiral Potts model; chiral clock model; star-triangle equations; Yang-Baxter equations; interfacial tension; wetting; superwetting;