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arXiv:0912.4549 · Journal of Physics A Mathematical and Theoretical

Spontaneous magnetization of the superintegrable chiral Potts model: calculation of the determinantDPQ

R. J. Baxter

Published Mar 16, 2010• 17 citations
arXivHTMLPDF

Abstract

For the Ising model, the calculation of the spontaneous magnetization leads to the problem of evaluating a determinant. Yang did this by calculating the eigenvalues in the large-lattice limit. Montroll, Potts and Ward expressed it as a Toeplitz determinant and used Szegő's theorem: this is almost certainly the route originally travelled by Onsager. For the corresponding problem in the superintegrable chiral Potts model, neither approach appears to work: here we show that the determinant D PQ can be expressed as that of a product of two Cauchy-like matrices. One can then use the elementary exact formula for the Cauchy determinant. One of course regains the known result, originally conjectured in 1989.

DOI
10.1088/1751-8113/43/14/145002

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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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We study in detail the asymptotic behavior, for large separations, of the correlation between two spins in the case of the two-dimensional Ising model without magnetic field. In the limit of infinite separation, this correlation is equal to the square of the spontaneous magnetization per spin. This paper is devoted to answering the question how, for fixed temperature, the correlation approaches this limiting value. The investigation is restricted to the situation where the two spins under consideration lie on the same row of the two-dimensional lattice. There are three distinct cases where the fixed temperature is above, below, or equal to the critical temperature or the Curie temperature. In the first two cases, the limiting value is approached exponentially as a function of the separation, while at the critical temperature the correlation behaves asymptotically as the inverse fourth root of the separation. In this paper, we evaluate exactly the first four terms of the asymptotic expansion in the first case, the first three terms in the second case, and the coefficients of the terms proportional to the -\textonequarter{} and the $\ensuremath{-}\frac{9}{4}$ powers of the separation in the third case. All these results are obtained by first expressing the correlation as a Toeplitz determinant, and the method used is capable of giving in principle the entire asymptotic expansion for large classes of such determinants, when the size of the determinant approaches infinity. The explicit results for the two-dimensional Ising model also serve as an example where the prescription of summing the leading terms, or the most divergent terms, fails.

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For the Ising model, the calculation of the spontaneous magnetization leads to the problem of evaluating a determinant. Yang did this by calculating the eigenvalues in the large-lattice limit. Montroll, Potts and Ward expressed it as a Toeplitz determinant and used Szegő's theorem: this is almost certainly the route originally travelled by Onsager. For the corresponding problem in the superintegrable chiral Potts model, neither approach appears to work: here we show that the determinant D PQ can be expressed as that of a product of two Cauchy-like matrices. One can then use the elementary exact formula for the Cauchy determinant. One of course regains the known result, originally conjectured in 1989.
We adapt our previous results for the “partition function ” of the superintegrable chiral Potts model with open boundaries to obtain the corresponding matrix elements of e −αH, where H is the associated hamiltonian. The spontaneous magnetization Mr can be expressed in terms of particular matrix elements of e −αH S r 1 e−βH, where S1 is a diagonal matrix. We present a conjecture for these matrix elements as an m by m determinant, where m is proportional to the width of the lattice. The author has previously derived the spontaneous magnetization of the chiral Potts model by analytic means, but hopes that this work will facilitate a more algebraic derivation, similar to that of Yang for the Ising model. KEY WORDS: Statistical mechanics, lattice models, transfer matrices. 1
We show how Z -invariance in the chiral Potts model provides a strategy to calculate the pair correlation in the general integrable chiral Potts model using only the superintegrable eigenvectors. When the distance between the two spins in the correlation function becomes infinite it becomes the square of the order parameter. In this way, we show that the spontaneous magnetization can be expressed in terms of the inner products of the eigenvectors of the N asymptotically degenerate maximum eigenvalues. Using our previous results on these eigenvectors, we are able to obtain the order parameter as a sum almost identical to the one given by Baxter. This gives the known spontaneous magnetization of the chiral Potts model by an entirely different approach.
In this paper we rederive, by simpler methods, the Onsager-Kaufman formulas for the correlations and the Onsager formula for the spontaneous magnetization of the rectangular two-dimensional Ising lattice. The Pfaffian approach is used to derive the correlations in terms of Pfaffians, and for the correlations in a row a single Toeplitz determinant is obtained which is proved equivalent to the Onsager-Kaufman result. The spontaneous magnetization is obtained as the limiting value of an infinite Toeplitz determinant and its evaluation is facilitated by use of the generalization of a result first published by Szego.
We conjecture exact and simple formulas for some physical quantities in two quantum chains. A classic result of this type is Onsager, Kaufman and Yang's formula for the spontaneous magnetization in the Ising model, subsequently generalized to the chiral Potts models. We conjecture that analogous results occur in the XYZ chain when the couplings obey J x J y + J y J z + J x J z = 0, and in a related fermion chain with strong interactions and supersymmetry. We find exact formulas for the magnetization and gap in the former, and the staggered density in the latter, by exploiting the fact that certain quantities are independent of finite-size effects.
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 study in detail the asymptotic behavior, for large separations, of the correlation between two spins in the case of the two-dimensional Ising model without magnetic field. In the limit of infinite separation, this correlation is equal to the square of the spontaneous magnetization per spin. This paper is devoted to answering the question how, for fixed temperature, the correlation approaches this limiting value. The investigation is restricted to the situation where the two spins under consideration lie on the same row of the two-dimensional lattice. There are three distinct cases where the fixed temperature is above, below, or equal to the critical temperature or the Curie temperature. In the first two cases, the limiting value is approached exponentially as a function of the separation, while at the critical temperature the correlation behaves asymptotically as the inverse fourth root of the separation. In this paper, we evaluate exactly the first four terms of the asymptotic expansion in the first case, the first three terms in the second case, and the coefficients of the terms proportional to the -\textonequarter{} and the $\ensuremath{-}\frac{9}{4}$ powers of the separation in the third case. All these results are obtained by first expressing the correlation as a Toeplitz determinant, and the method used is capable of giving in principle the entire asymptotic expansion for large classes of such determinants, when the size of the determinant approaches infinity. The explicit results for the two-dimensional Ising model also serve as an example where the prescription of summing the leading terms, or the most divergent terms, fails.