Chiral Potts 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.S. Howes, L.P. Kadanoff and M. den Nijs (1983), Nuclear Physics B 215, 169. For N = 3 and 4, the integrable case was discovered in 1986 in Stony Brook and published the following year.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.

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.{{cite journal |last1=Fateev |first1=V. A.

|last2=Zamolodchikov |first2=A. B.

|title=Self-dual solutions of the star-triangle relations in ZN-models

|journal=Physics Letters A |date=18 October 1982

|volume=92 |issue=1 |pages=37–39 |doi=10.1016/0375-9601(82)90736-8

|url=https://www.sciencedirect.com/science/article/pii/0375960182907368

|access-date=11 July 2023 |language=en |issn=0375-9601|url-access=subscription }}

By removing certain restrictions of the work of Alcaraz and Santos,{{cite journal

|last1=Alcaraz |first1=Francisco C.

|last2=Lima Santos |first2=A.

|title=Conservation laws for Z(N) symmetric quantum spin models and their exact ground state energies

|journal=Nuclear Physics B |date=24 November 1986

|volume=275 |issue=3 |pages=436–458 |doi=10.1016/0550-3213(86)90608-5

|url=https://www.sciencedirect.com/science/article/pii/0550321386906085

|language=en |issn=0550-3213|url-access=subscription}} a more general self-dual case of the integrable chiral Potts model was discovered. The weight are given in product formH. 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.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. and the parameters in the weight are shown to be on the Fermat curve, with genus greater than 1.

The general solution for all k (the temperature variable) was found. 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.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

From the seriesM. 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) the order parameter was conjecturedAlbertini 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 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 2005Baxter R. J. (2005),

"Derivation of the order parameter of the chiral Potts model",

Physical Review Letters, 94 130602 (3 pp) arXiv:cond-mat/0501227.Baxter R. J. (2005),

"The order parameter of the chiral Potts model",

Journal of Statistical Physics 120, 1–36: arXiv:cond-mat/0501226. using functional equations and the "broken rapidity line" technique of Jimbo et al.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: arXiv:hep-th/9211066. assuming two mild analyticity

conditions of the type commonly used in the field of Yang–Baxter integrable models. Most recently, in a series of papersBaxter R. J. (2008)

"Algebraic reduction of the Ising model",

Journal of Statistical Physics 132, 959–82, arXiv:0803.4036;Baxter R. J. (2008),

"A conjecture for the superintegrable chiral Potts model",

Journal of Statistical Physics 132, 983–1000, arXiv:0803.4037;Baxter R J (2009),

"Some remarks on a generalization of the superintegrable chiral Potts model",

Journal of Statistical Physics 137, 798–813, arXiv:0906.3551;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) arXiv:0912.4549.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

arXiv:1001.0281.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 arXiv:0912.5027.Au-Yang H and Perk J. H. H. (2011),

"Spontaneous magnetization of the integrable chiral Potts model",

Journal of Physics A 44, 445005 (20pp), arXiv:1003.4805.

an algebraic (Ising-like) way of obtaining the order parameter has been given, giving more insight into the algebraic structure.

In 1990 Bazhanov and StroganovV. 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. showed that there exist L-operators (Lax operator) which satisfy the Yang–Baxter equation

:

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 as

:

This inspired a breakthrough, namely the functional relations for the transfer matrices of the chiral Potts models were discovered.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.

Using these functional relations, Baxter was able to calculate the eigenvalues of the transfer matrix of the chiral Potts model,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. 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.Baxter R. J. (1993), "Chiral Potts model with skewed boundary conditions", Journal of Statistical Physics 73, 461–95.Baxter R. J. (1994), "Interfacial tension of the chiral Potts model", Journal of Physics A 27, pp 1837–49.

The integrable chiral Potts weights are given in product form as

:$$

W_{pq}(n)\!=\!\Big({\mu_p\over\mu_q}\Big)^{\!\!n}\prod_{j=1}^n

{y_q-x_p\omega^j\over y_p-x_q\omega^j},\quad

\overline W_{pq}(n)\!=\!\big({\mu_p\mu_q}\big)^{\!n}\!\prod_{j=1}^n

{\omega x_p\!-\!x_q\omega^j\over y_q\!-\!y_p\omega^j},

where $\backslash omega^N\; =\; 1$ is a primitive root of unity and we associate with each rapidity variable p three variables $(x\_p,\; y\_p,\; \backslash mu\_p)$ satisfying

:$x\_p^N+y\_p^N=k(1+x\_p^N\; y\_p^N),\backslash quad$

k'^2+k^2=1,\quad

k'\mu_p^N=1-k y_p^N,\quad

k'\mu_p^{-N}=1-k x_p^N.

It is easy to see that

:$$

W_{pp}(a-b)=1,\quad \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,\quad

\sum_{d=0}^{N-1}\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.

Au-Yang Helen, Perk H. H. Jacques (2016), arXiv:1601.01014

{{multiple image

| align = center

| image1 = Reidemeister Move I.png

| width1 = 300

| alt1 = Property of the Weights: Reidemeister Move I

| caption1 = Property of the Weights: Reidemeister Move I

| image2 = Reidemeister Move II.png

| width2 = 380

| alt2 = Inversion Relation of the Weights: Reidemeister Move II

| caption2 = Inversion Relation of the Weights: Reidemeister Move II

| image3 = Reidemeister Move III.png

| width3 = 598

| alt3 = Star-triangle relation: Reidemeister Move III

| caption3 = Star-triangle relation: Reidemeister Move III

}}

File:Weights of The Chiral Potts Model.png

{{clear}}

• Z N model

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Category:Lattice models

Category:Spin models

Category:Statistical mechanics