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Physics Today

Sorting Out the Potts Models

R. J. Baxter

Published Dec 1, 2005• 4 citations
Publisher

Abstract

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.

DOI
10.1063/1.2169427

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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;

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For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent $2-\sqrt{2} \approx 0.5858,$ comparable to the exponent $4/7 \approx 0.5714$ found from renormalization arguments.

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Bohr’s molecular model and the melding of classical and quantum mechanics

Anatoly A. Svidzinsky, Marlan O. Scully, Dudley R. Herschbach

Svidzinsky, Scully, and Herschbach reply: We agree with M. Y. Amusia that a statement we made about the infinite dimension limit is “lacking in rigor.” It was intended simply to provide heuristic insight into why, for a wide variety of problems, the large-D limit has proven to be a useful starting approximation to obtain results for D = 3.Amusia’s other concerns are answered in a 1975 paper on a D-scaling treatment of helium by David Herrick and Frank Stillinger.11. D. R. Herrick, F. H. Stillinger, Phys. Rev. A 11, 42 (1975). https://doi.org/10.1103/PhysRevA.11.42 They gave a rigorous derivation of the D-dimensional hydrogen atom Hamiltonian, shown as equation 1 in Amusia’s letter. That solution applies for D ≥ 2, as implied in the figure included in box 1 of our article. Herrick and Stillinger also showed that for the correct D → 1 limit, the Z/r term becomes a δ-function. In D-scaling, contrary to Amusia’s assumption, the D = 3 form of Coulomb’s law can be used for dimensional continuation to the large-D limit. The D-dependent similarity transformation affects the Laplacian, not the potential energy. Both theory and application are amply presented in references given in our article (particularly references 8–11). D-scaling, using just elementary algebra, has attained correlation energies for multielectron atoms with accuracy comparable to or better than conventional electronic calculations.22. See, for example, S. Kais, R. Bleil, J. Chem. Phys. 102, 7472 (1995); https://doi.org/10.1063/1.469059D. K. Watson, D. Z. Goodson, Phys. Rev. A 51, R5 (1995), and references therein. https://doi.org/10.1103/PhysRevA.51.R5The papers of Leonard Mlodinow33. See D. R. Herschbach, J. Chem. Phys. 84, 838 (1986), for analysis of the papers cited in reference 1 of Mlodinow’s letter. https://doi.org/10.1063/1.450584 and several other authors, especially the earlier paper by Herrick and Stillinger,11. D. R. Herrick, F. H. Stillinger, Phys. Rev. A 11, 42 (1975). https://doi.org/10.1103/PhysRevA.11.42 fostered the development of D-scaling for electronic structure. Since the kinship of Bohr’s model to dimensional scaling was not recognized until 2005, we did not dwell on that history, other than citing the tutorial article by Edward Witten. The treatment of H2 that Mlodinow cites in the second part of his reference 3 is a deliberately drastic approximation and gives less than 40% of the bond dissociation energy.As emphasized by Petar Grujic, the bold, perplexing enterprise by Niels Bohr motivated much further work melding classical and quantum mechanics. We note that D-scaling has a distinctive character. It might aptly be termed “semiquantum” rather than semiclassical. Although in the large-D limit the equations become classical, quantum mechanics is hidden in the D-dependent units adopted for distance and energy. At that limit, electrons take fixed positions in the D-scaled space and the first-order correction in (1/D) has them execute harmonic vibrations about those positions. In the prequantum era, such behavior was postulated by Gilbert Lewis and Irving Langmuir,44. P. Coffey, Cathedrals of Science: The Personalities and Rivalries That Made Modern Chemistry, Oxford U. Press, New York (2008), chap. 5, esp. p. 139. motivated by chemical arguments but disdained by physicists and considered incompatible with the Bohr model.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. D. R. Herrick, F. H. Stillinger, Phys. Rev. A 11, 42 (1975). https://doi.org/10.1103/PhysRevA.11.42, Google ScholarCrossref, ISI2. See, for example, S. Kais, R. Bleil, J. Chem. Phys. 102, 7472 (1995); https://doi.org/10.1063/1.469059, Google ScholarCrossref, ISID. K. Watson, D. Z. Goodson, Phys. Rev. A 51, R5 (1995), and references therein. https://doi.org/10.1103/PhysRevA.51.R5, , Google ScholarCrossref, ISI3. See D. R. Herschbach, J. Chem. Phys. 84, 838 (1986), for analysis of the papers cited in reference 1 of Mlodinow’s letter. https://doi.org/10.1063/1.450584, Google ScholarCrossref, ISI4. P. Coffey, Cathedrals of Science: The Personalities and Rivalries That Made Modern Chemistry, Oxford U. Press, New York (2008), chap. 5, esp. p. 139. Google Scholar© 2014 American Institute of Physics.

Physics Today•Aug 1, 2014•2 citations
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.
Exact Spectrum of the Critical Three-State Potts Chain (G Albertini) New Results Related to the Chiral Potts Model (H Au-Yang & J H H Perk) Chiral Potts Model: Corner Transfer Matrices and Parametrizations (R J Baxter) Solvable Lattice Models in Three Dimensions (VV Bazhanov) Dressing Symmetries (D Bernard) Knot Theory and Chem-Simons Lagrangian: Some Remarks about the corresponding Diagrams (P Cartier) Recent Progress in 2-D Quantum Gravity (V Dotsenko) On Coordinate Algebras of Quantum Groups at Root of Unity (B Enriquez) Integrability in Discrete Systems (B Grammaticos) New Construction of Solvable RSOS Models (S 0 Warnaar) Birational Automorphisms and Yang-Baxter Equations (J-M Maillard) Yang-Baxter Equations and Non-Commutative Differential Calculus (G Maltsiniotis) Physics Beyond Quasi-Particules: Spectrum and Completeness of the Three-State Superintegrable Chiral Potts Model (B M McCoy) Analytic Calulation of Conformal Weights for A-D-E Lattice Models (P Pearce) Quantum Tops (A Reiman) Remarks on Quasi-Hopf Algebras (M Rosso) Symmetries of the Yang-Baxter Equations: A Remarkable Family of Discrete Groups (C-M Viallet) The Yang-Baxter Equation in Knot Theory (F Y Wu).
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;
For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent 2−2≈0.5858,2-\sqrt{2} \approx 0.5858,2−2​≈0.5858, comparable to the exponent 4/7≈0.57144/7 \approx 0.57144/7≈0.5714 found from renormalization arguments.
Svidzinsky, Scully, and Herschbach reply: We agree with M. Y. Amusia that a statement we made about the infinite dimension limit is “lacking in rigor.” It was intended simply to provide heuristic insight into why, for a wide variety of problems, the large-D limit has proven to be a useful starting approximation to obtain results for D = 3.Amusia’s other concerns are answered in a 1975 paper on a D-scaling treatment of helium by David Herrick and Frank Stillinger.11. D. R. Herrick, F. H. Stillinger, Phys. Rev. A 11, 42 (1975). https://doi.org/10.1103/PhysRevA.11.42 They gave a rigorous derivation of the D-dimensional hydrogen atom Hamiltonian, shown as equation 1 in Amusia’s letter. That solution applies for D ≥ 2, as implied in the figure included in box 1 of our article. Herrick and Stillinger also showed that for the correct D → 1 limit, the Z/r term becomes a δ-function. In D-scaling, contrary to Amusia’s assumption, the D = 3 form of Coulomb’s law can be used for dimensional continuation to the large-D limit. The D-dependent similarity transformation affects the Laplacian, not the potential energy. Both theory and application are amply presented in references given in our article (particularly references 8–11). D-scaling, using just elementary algebra, has attained correlation energies for multielectron atoms with accuracy comparable to or better than conventional electronic calculations.22. See, for example, S. Kais, R. Bleil, J. Chem. Phys. 102, 7472 (1995); https://doi.org/10.1063/1.469059D. K. Watson, D. Z. Goodson, Phys. Rev. A 51, R5 (1995), and references therein. https://doi.org/10.1103/PhysRevA.51.R5The papers of Leonard Mlodinow33. See D. R. Herschbach, J. Chem. Phys. 84, 838 (1986), for analysis of the papers cited in reference 1 of Mlodinow’s letter. https://doi.org/10.1063/1.450584 and several other authors, especially the earlier paper by Herrick and Stillinger,11. D. R. Herrick, F. H. Stillinger, Phys. Rev. A 11, 42 (1975). https://doi.org/10.1103/PhysRevA.11.42 fostered the development of D-scaling for electronic structure. Since the kinship of Bohr’s model to dimensional scaling was not recognized until 2005, we did not dwell on that history, other than citing the tutorial article by Edward Witten. The treatment of H2 that Mlodinow cites in the second part of his reference 3 is a deliberately drastic approximation and gives less than 40% of the bond dissociation energy.As emphasized by Petar Grujic, the bold, perplexing enterprise by Niels Bohr motivated much further work melding classical and quantum mechanics. We note that D-scaling has a distinctive character. It might aptly be termed “semiquantum” rather than semiclassical. Although in the large-D limit the equations become classical, quantum mechanics is hidden in the D-dependent units adopted for distance and energy. At that limit, electrons take fixed positions in the D-scaled space and the first-order correction in (1/D) has them execute harmonic vibrations about those positions. In the prequantum era, such behavior was postulated by Gilbert Lewis and Irving Langmuir,44. P. Coffey, Cathedrals of Science: The Personalities and Rivalries That Made Modern Chemistry, Oxford U. Press, New York (2008), chap. 5, esp. p. 139. motivated by chemical arguments but disdained by physicists and considered incompatible with the Bohr model.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. D. R. Herrick, F. H. Stillinger, Phys. Rev. A 11, 42 (1975). https://doi.org/10.1103/PhysRevA.11.42, Google ScholarCrossref, ISI2. See, for example, S. Kais, R. Bleil, J. Chem. Phys. 102, 7472 (1995); https://doi.org/10.1063/1.469059, Google ScholarCrossref, ISID. K. Watson, D. Z. Goodson, Phys. Rev. A 51, R5 (1995), and references therein. https://doi.org/10.1103/PhysRevA.51.R5, , Google ScholarCrossref, ISI3. See D. R. Herschbach, J. Chem. Phys. 84, 838 (1986), for analysis of the papers cited in reference 1 of Mlodinow’s letter. https://doi.org/10.1063/1.450584, Google ScholarCrossref, ISI4. P. Coffey, Cathedrals of Science: The Personalities and Rivalries That Made Modern Chemistry, Oxford U. Press, New York (2008), chap. 5, esp. p. 139. Google Scholar© 2014 American Institute of Physics.