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The inversion relation method for obtaining the free energy of the chiral Potts model

R. J. Baxter

Published Jan 1, 2008• 11 citations
OpenAlex

Abstract

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.

Related papers

The challenge of the chiral Potts model

R. J. Baxter

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

•Jan 1, 2005•6 citations

The challenge of the chiral Potts model

R. J. Baxter

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.

arXiv:cond-mat/0510683•Journal of Physics Conference Series•Jun 1, 2006•9 citations

Boundary conditions and the free energy for statistical mechanical models

Paul Martin

The author shows that the choice of boundary conditions can significantly affect the calculation of the free energy for discrete statistical mechanical models on finite or semi-infinite lattices when the exponentiated coupling is extended into the complex plane. He shows however, that the boundary conditions most consistent with boundary configuration independence in the direction in which a lattice is infinite give a unique analytic function for the free energy. This unique function is the free energy most appropriate for extrapolation to the full thermodynamic limit. The author shows that such boundary conditions are orthogonal to all but one degenerate ferromagnetic ground states at zero temperature.

Journal of Physics A Mathematical and General•Jun 21, 1987•9 citations

Sorting Out the Potts Models

R. J. Baxter

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.

Physics Today•Dec 1, 2005•4 citations

A STEP-BY-STEP APPROACH TO INTEGRABILITY

David Hansel, J.-M. Maillard, Pál Ruján

We try to elucidate the role played by different symmetries in simple models of statistical mechanics. Starting with obvious symmetries for the partition function and combining them with duality relations we obtain a set of constraints on the possible algebraic varieties relevant for the integrable manifolds and the phase diagram of the lattice model. When imposing in addition the inversion relation special polynomials are obtained, which are close and sometimes identical to the set of equations defining the parameter subspace of the integrable models. Our procedure is detailed on the q-state chiral Potts models on a square lattice, in particular for q=3 and 4.

International Journal of Modern Physics B•Oct 1, 1989•5 citations

ZERO-TEMPERATURE SKEWED CHIRAL POTTS MODEL

R. J. Baxter

Functional relations have previously been obtained for the eigenvalues of the transfer matrices of the chiral Potts model. Introducing skewed boundary conditions is equivalent to merely modifying the quantum number of the spin shift operator in the relations (which accounts for at least some of the previously noted "spurious solutions"). As a first step towards calculating the general interfacial tension, we consider the model in a zero-temperature limit. It is still non-trivial, there being near-vertical free interfaces separating domains of different spin value. These interfaces behave like the "lines of down arrows" in the ice model, so one may hope to follow Lieb and use the Bethe ansatz to evaluate the partition function. It turns out that this can indeed be done. There is no wetting of an interface by intermediate phases.

Reviews in Mathematical Physics•Jan 1, 1994•1 citation
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.
The author shows that the choice of boundary conditions can significantly affect the calculation of the free energy for discrete statistical mechanical models on finite or semi-infinite lattices when the exponentiated coupling is extended into the complex plane. He shows however, that the boundary conditions most consistent with boundary configuration independence in the direction in which a lattice is infinite give a unique analytic function for the free energy. This unique function is the free energy most appropriate for extrapolation to the full thermodynamic limit. The author shows that such boundary conditions are orthogonal to all but one degenerate ferromagnetic ground states at zero temperature.
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.
We try to elucidate the role played by different symmetries in simple models of statistical mechanics. Starting with obvious symmetries for the partition function and combining them with duality relations we obtain a set of constraints on the possible algebraic varieties relevant for the integrable manifolds and the phase diagram of the lattice model. When imposing in addition the inversion relation special polynomials are obtained, which are close and sometimes identical to the set of equations defining the parameter subspace of the integrable models. Our procedure is detailed on the q-state chiral Potts models on a square lattice, in particular for q=3 and 4.
Functional relations have previously been obtained for the eigenvalues of the transfer matrices of the chiral Potts model. Introducing skewed boundary conditions is equivalent to merely modifying the quantum number of the spin shift operator in the relations (which accounts for at least some of the previously noted "spurious solutions"). As a first step towards calculating the general interfacial tension, we consider the model in a zero-temperature limit. It is still non-trivial, there being near-vertical free interfaces separating domains of different spin value. These interfaces behave like the "lines of down arrows" in the ice model, so one may hope to follow Lieb and use the Bethe ansatz to evaluate the partition function. It turns out that this can indeed be done. There is no wetting of an interface by intermediate phases.