Chiral Potts model explained

The chiral Potts model is a spin model on a planar lattice in statistical mechanics studied by Helen Au-Yang Perk and Jacques Perk, among others. It may be viewed as a generalization of the Potts model, and as with the Potts model, the model is defined by configurations which are assignments of spins to each vertex of a graph, where each spin can take one of

N values. To each edge joining vertices with assigned spins n and n' , a Boltzmann weight W(n,n') is assigned. For this model, chiral means that W(n,n') ≠ W(n',n) . When the weights satisfy the Yang–Baxter equation, it is integrable, in the sense that certain quantities can be exactly evaluated.

For the integrable chiral Potts model, the weights are defined by a high genus curve, the chiral Potts curve.^{[1] [2]} Unlike the other solvable models,^{[3] [4]} whose weights are parametrized by curves of genus less or equal to one, so that they can be expressed in terms of trigonometric functions, rational functions for the genus zero case, or by theta functions for the genus 1 case, this model involves high genus theta functions, for which the theory is less well-developed.

The related chiral clock model, which was introduced in the 1980s by David Huse and Stellan Ostlund independently, is not exactly solvable, in contrast to the chiral Potts model.

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.^[5] For N = 3 and 4, the integrable case was discovered in 1986 in Stony Brook and published the following year.^{[1] [6]}

Self-dual case

The model is called self-dual if the Fourier transform of the weight function returns the same function. A special (genus 1) case had been solved in 1982 by Fateev and Zamolodchikov.^[7] By removing certain restrictions of the work of Alcaraz and Santos,^[8] a more general self-dual case of the integrable chiral Potts model was discovered.^[1] The weight are given in product form^{[9] [10]} and the parameters in the weight are shown to be on the Fermat curve, with genus greater than 1.

General case

The general solution for all k (the temperature variable) was found.^[2] The weights were also given in product form and it was tested computationally (on Fortran) that they satisfy the star–triangle relation. The proof was published later.^[11]

Results

Order parameter

From the series^{[5] [12]} the order parameter was conjectured^[13] to have the simple form$$\backslash langle\; \backslash sigma^n\backslash rangle=(1-k\text{'}^2)^\backslash beta,\backslash quad\; \backslash beta=n(N-n)/2N^2.$$It took many years to prove this conjecture, as the usual corner transfer matrix technique could not be used, because of the higher genus curve. This conjecture was proven by Baxter in 2005^{[14] [15]} using functional equations and the "broken rapidity line" technique of Jimbo et al.^[16] assuming two mild analyticityconditions of the type commonly used in the field of Yang–Baxter integrable models. Most recently, in a series of papers^{[17] [18] [19] [20] [21] [22] [23]} an algebraic (Ising-like) way of obtaining the order parameter has been given, giving more insight into the algebraic structure.

Connection to six vertex model

In 1990 Bazhanov and Stroganov^[24] showed that there exist L-operators (Lax operator) which satisfy the Yang–Baxter equation

L ^{j _{1\beta}} _{i _{1\alpha}} (x)L ^{j _{2\gamma}} _{i _{2\beta}} (y)R ^{k _{1k} _{2}} _{j _{1j} _{2}} (y/x) = R ^{j _{1j} _{2}} _{i _{1i} _{2}} (y/x)L ^{k _{2\beta}} _{j _{2\alpha}} (y)L ^{k _{1\gamma}} _{j _{1\beta}} (x), 0 <i,j,k\le 1 , 0 \le \alpha, \beta,\gamma\le N -1 . where the 2 × 2 R-operator (R-matrix) is the six vertex model R-matrix (see Vertex model). The product of four chiral Potts weight S was shown to intertwine two L-operators asL ^{i _{2\alpha} _{2}} _{i _{1\alpha} _{1}} {\hat L} ^{i _{3\beta} _{2}} _{i _{2\beta} _{1}} S ^{\alpha _{3\beta} _{3}} _{\alpha _{2\beta} _{2}} = S ^{\alpha _{2\beta} _{2}} _{\alpha _{1\beta} _{1}} {\hat L} ^{i _{2\beta} _{3}} _{i _{1\beta} _{2}} L ^{i _{3\alpha} _{3}} _{i _{2\alpha} _{2}} , 0 <i _{i\le1,} 0 \le \alpha _{i,} \beta _{i\le} N -1 . This inspired a breakthrough, namely the functional relations for the transfer matrices of the chiral Potts models were discovered.^[25]

Free energy and interfacial tension

Using these functional relations, Baxter was able to calculate the eigenvalues of the transfer matrix of the chiral Potts model,^[26] and obtained the critical exponent for the specific heat α=1-2/N, which was also conjectured in reference 12. The interfacial tension was also calculated by him with the exponent μ=1/2+1/N.^{[27] [28]}

Relation with knot theory

The integrable chiral Potts weights are given in product form ^[2] as

W _{pq} (n) = ( {\mu _{p\over\mu} ^{n} ^{q} )} \prod ^{n {y} _{q - x} ^{j\over} ^{p\omega} y _{p-x} ^{j},} ^{q\omega} \overline W _{pq} (n) = ( {\mu _{p\mu} ^{n} ^{q} )} \prod ^{n {\omega} _{j =1} x _{p-x} ^{j\over} ^{q\omega} y _{q-y} ^{j},} ^{p\omega} where \omega ^{N} = 1 is a primitive root of unity and we associate with each rapidity variable p three variables (x _{p,} y _{p,} \mu _{p)} satisfyingx ^{N} _{p} y ^{N), k'} _{p} ^{2+k} ^{N =1- k} ^{p} y ^{- N} _{p} =1- k x ^{N.} _{p} It is easy to see thatW _{pp} (a - b) =1 , \overline W _{pp} (a - b) = \delta _{a,b} which is similar to Reidemeister move I. It was also known that the weights satisfy the inversion relation,W _{pq} (a - b)W _{qp} (a - b) =1 , \sum ^{N -1} _{d =0} \overline W _{pq} (a - d)\overline W _{qp} (d - a') = r _{pq} \delta _{a,a'} . This is equivalent to Reidemeister move II. The star-triangle relation\sum ^{N} _{d =1} {\overline W} _{pr} (a - d)W _{pq} (d - c){\overline W} _{rq} (d - b) = R _{pqr} \overline W _{pq} (a - b){ W} _{pr} (b - c)W _{rq} (a - c) is equivalent to Reidemeister move III. These are shown in the figures below.^[29]

See also

• Z N model

Notes and References

1. Au-Yang . Helen . McCoy . Barry M. . Perk . Jacques H. H. . Tang . Shuang . Yan . Mu-Lin . Commuting transfer matrices in the chiral Potts models: Solutions of star-triangle equations with genus>1 . Physics Letters A . 10 August 1987 . 123 . 5 . 219–223 . 10.1016/0375-9601(87)90065-X . en . 0375-9601.
2. Baxter . R. J. . Perk . J. H. H. . Au-Yang . H. . New solutions of the star-triangle relations for the chiral potts model . Physics Letters A . 28 March 1988 . 128 . 3 . 138–142 . 10.1016/0375-9601(88)90896-1 . 10 July 2023 . en . 0375-9601.
3. Book: Baxter . Rodney J. . Exactly solved models in statistical mechanics . 2007 . Dover Publications, Inc . Mineola, N.Y . 978-0486462714.
4. Book: McCoy . Barry M. . Advanced statistical mechanics . 2010 . Oxford university press . Oxford . 978-0199556632.
5. S. Howes, L.P. Kadanoff and M. den Nijs (1983), Nuclear Physics B 215, 169.
6. 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.
7. Fateev . V. A. . Zamolodchikov . A. B. . Self-dual solutions of the star-triangle relations in ZN-models . Physics Letters A . 18 October 1982 . 92 . 1 . 37–39 . 10.1016/0375-9601(82)90736-8 . 11 July 2023 . en . 0375-9601.
8. Alcaraz . Francisco C. . Lima Santos . A. . Conservation laws for Z(N) symmetric quantum spin models and their exact ground state energies . Nuclear Physics B . 24 November 1986 . 275 . 3 . 436–458 . 10.1016/0550-3213(86)90608-5 . en . 0550-3213.
9. H. Au-Yang, B. M. McCoy, J. H. H. Perk, and S. Tang (1988), "Solvable models in statistical mechanics and Riemann surfaces of genus greater than one", in Algebraic Analysis, Vol. 1, M. Kashiwara and T. Kawai, eds., Academic Press, pp. 29–40.
10. J.H.H. Perk (1987), "Star-triangle equations, quantum Lax pairs, and higher genus curves", in Proc. 1987 Summer Research Institute on Theta Functions, Proc. Symp. Pure Math., Vol. 49, part 1 (Am. Math. Soc., Providence, R.I., 1989), pp. 341–354.
11. Au-Yang H and Perk J H H (1989). "Onsager's star-triangle equation: Master key to integrability", Proc. Taniguchi Symposium, Kyoto, October 1988, Advanced Studies in Pure Mathematics vol 19 (Tokyo: Kinokuniya–Academic) pp 57–94
12. M. Henkel and J. Lacki, preprint Bonn-HE-85–22 and "Integrable chiral $Z_n$ quantum chains and a new class of trigonometric sums", Phys. Lett. 138A 105 (1989)
13. Albertini G., McCoy B. M., Perk J. H. H. and Tang S. (1989), "Excitation spectrum and order parameter for the integrable N-state chiral Potts model", Nuclear Physics B 314, 741–763
14. Baxter R. J. (2005), "Derivation of the order parameter of the chiral Potts model", Physical Review Letters, 94 130602 (3 pp) .
15. Baxter R. J. (2005), "The order parameter of the chiral Potts model", Journal of Statistical Physics 120, 1–36: .
16. Jimbo M., Miwa T. and Nakayashiki A. (1993), "Difference equations for the correlation functions of the eight-vertex model",Journal of Physics A

    Math. Gen. 26, 2199–210: .

17. Baxter R. J. (2008) "Algebraic reduction of the Ising model", Journal of Statistical Physics 132, 959–82, ;
18. Baxter R. J. (2008), "A conjecture for the superintegrable chiral Potts model",Journal of Statistical Physics 132, 983–1000, ;
19. Baxter R J (2009), "Some remarks on a generalization of the superintegrable chiral Potts model", Journal of Statistical Physics 137, 798–813, ;
20. Baxter R. J. (2010), "Spontaneous magnetization of the superintegrable chiral Potts model: calculation of the determinant D_PQ", Journal of Physics A 43, 145002 (16pp) .
21. Baxter R. J. (2010), "Proof of the determinantal form of the spontaneous magnetization of the superintegrable chiral Potts model", Australian & New Zealand Industrial and Applied Mathematics Journal, 51.
22. Iorgov N., Pakuliak S., Shadura V., Tykhyy Yu and von Gehlen G. (2009), "Spin operator matrix elements in the superintegrable chiral Potts quantum chain", Journal of Statistical Physics 139, 743–68 .
23. Au-Yang H and Perk J. H. H. (2011), "Spontaneous magnetization of the integrable chiral Potts model", Journal of Physics A 44, 445005 (20pp), .
24. V. V. Bazhanov and Yu. G. Stroganov (1990), "Chiral Potts model as a descendant of the six-vertex model", Journal of Statistical Physics 59, pp 799–817.
25. Baxter R. J., Bazhanov V. V. and Perk J. H. H. (1990), "Functional relations for transfer matrices of the chiral Potts model", International Journal of Modern Physics B 4, 803–70.
26. Baxter R J (1991), "Calculation of the eigenvalues of the transfer matrix of the chiral Potts model", Proceeding of Fourth Asia Pacific Physics Conference (Singapore: World Scientific) pp 42–58.
27. Baxter R. J. (1993), "Chiral Potts model with skewed boundary conditions", Journal of Statistical Physics 73, 461–95.
28. Baxter R. J. (1994), "Interfacial tension of the chiral Potts model", Journal of Physics A 27, pp 1837–49.
29. Au-Yang Helen, Perk H. H. Jacques (2016), arXiv:1601.01014