Abstract
An outstanding problem in statistical mechanics is the order parameter of the chiral Potts model. An elegant conjecture for this was made in 1983. It has since been successfully tested against series expansions, but there is as yet no proof of the conjecture. Here we show that if one makes a certain analyticity assumption similar to that used to derive the free energy, then one can indeed verify the conjecture. The method is based on the ‘‘broken rapidity line’’ approach pioneered by Jimbo et al. (J. Phys. A 26:2199--2210 (1993).).
Similar content being viewed by others
References
S. Howes L. P. Kadanoff M. den Nijs (1983) ArticleTitleQuantum model for commensurate-incommensurate transitions Nucl. Phys. B 215(FS7) 169–208 Occurrence Handle10.1016/0550-3213(83)90212-2
G. von Gehlen V. Rittenberg (1985) ArticleTitleZn-symmetric quantum chains with an infinite set of conserved charges and Zn zero modes Nucl. Phys. B 257(FS14) 351–370 Occurrence Handle10.1016/0550-3213(85)90350-5
H. Au-Yang B. M. McCoy J. H. H. Perk S. Tang M.-L. Yan (1987) ArticleTitleCommuting transfer matrices in the chiral Potts models: solutions of the star-triangle equations with genus >1 Phys. Lett. A 123 219–223 Occurrence Handle10.1016/0375-9601(87)90065-X
B. M. McCoy J. H. H. Perk S. Tang C.-H. Sah (1987) ArticleTitle{Commuting transfer matrices for the four-state self-dual chiral Potts model with a genus-three uniformizing Fermat curve} Phys. Lett. A 125 9–14 Occurrence Handle10.1016/0375-9601(87)90509-3
R. J. Baxter J. H. H. Perk H. Au-Yang (1988) ArticleTitle{New solutions of the star-triangle relations for the chiral Potts model} Phys. Lett. A 128 138–142 Occurrence Handle10.1016/0375-9601(88)90896-1
R. J. Baxter (1988) ArticleTitle{Free energy of the solvable chiral Potts model} J. Stat. Phys. 52 639–667 Occurrence Handle10.1007/BF01019722
V. V. Bazhanov (1990) ArticleTitleand Yu.G Stroganov, Chiral Potts model as a descendant of the six-vertex model, J. Stat. Phys. 59 799–817
R. J. Baxter V. V. Bazhanov J. H. H. Perk (1990) ArticleTitleFunctional relations for transfer matrices of the chiral Potts model Int. J. Mod. Phys. B 4 803–870 Occurrence Handle10.1142/S0217979290000395
R. J. Baxter (1990) ArticleTitleChiral Potts model: eigenvalues of the transfer matrix Phys. Lett. A 146 110–114 Occurrence Handle10.1016/0375-9601(90)90646-6
R. J. Baxter (1991) Calculation of the eigenvalues of the transfer matrix of the chiral Potts model S. H. Ahn Il-T. Cheon S. H. Choh C. Lee (Eds) Proc. Fourth Asia-Pacific Physics Conference, Vol. 1 World-Scientific Singapore 42–57
R. J. Baxter (1993) ArticleTitleCorner transfer matrices of the chiral Potts model II. The triangular lattice, J. Stat. Phys. 70 535–582
C. N. Yang (1952) ArticleTitleThe spontaneous magnetization of a two-dimensional Ising model Phys. Rev 85 808–816 Occurrence Handle10.1103/PhysRev.85.808
L. Onsager (1971) The Ising model in two dimensions R. E. Mills E. Ascher R. I. Jaffee (Eds) Critical Phenomena in Alloys, Magnets and Superconductors McGraw-Hill NY 3–12
M. Henkel and J. Lacki, preprint Bonn-HE-85-22 (1985).
G. Albertini B. M. McCoy J. H. H. Perk S. Tang (1989) ArticleTitleExcitation spectrum and order parameter for the integrable N-state chiral Potts model Nuc. Phys. B 314 741–763 Occurrence Handle10.1016/0550-3213(89)90415-X
M. Henkel J. Lacki (1989) ArticleTitleIntegrable chiral Zn quantum chains and a new class of trigonometric sums Phys. Lett. A 138 105–109 Occurrence Handle10.1016/0375-9601(89)90872-4
C. N. Yang Selected Papers 1945–1980 (1983) with Commentary W. H. Freeman and Co, San Francisco
R. J. Baxter (1982) Exactly Solved Models in Statistical Mechanics Academic London
M. Jimbo T. Miwa A. Nakayashiki (1993) ArticleTitleDifference equations for the correlation functions of the eight-vertex model J. Phys. A 26 2199–2210
R. J. Baxter (1998) ArticleTitleFunctional relations for the order parameters of the chiral Potts model J. Stat. Phys. 91 499–524 Occurrence Handle10.1023/A:1023096408679
R. J. Baxter (1982) ArticleTitleThe inversion relation method for some two-dimensional exactly solved models in lattice statistics J. Stat. Phys. 28 1–41 Occurrence Handle10.1007/BF01011621
R. J. Baxter (2003) ArticleTitleThe inversion relation method for obtaining the free energy of the chiral Potts model Physica A 322 407–431
R. J. Baxter (2004) ArticleTitleTransfer matrix functional relations for the generalized τ 2(tq) model J. Stat. Phys. 117 1–25 Occurrence Handle10.1023/B:JOSS.0000044062.64287.b9 Occurrence HandleMR2098556
R. J. Baxter (1989) ArticleTitleSuperintegrable chiral Potts model: thermodynamic properties an inverse model, and a simple associated hamiltonian, J. Stat. Phys. 57 1–39
R. J. Baxter (1978) ArticleTitleSolvable eight-vertex model on an arbitrary planar lattice Phil Trans. Roy. Soc. 289 315–346
R. J. Baxter (1998) ArticleTitleFunctional relations for the order parameters of the chiral Potts model: low-temperature expansions Physica A 260 117–130
R. J. Baxter (2003) ArticleTitleThe Riemann surface of the chiral Potts free energy function J. Stat. Phys. 112 1–26 Occurrence Handle10.1023/A:1023611702183
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Baxter, R. The Order Parameter of the Chiral Potts Model. J Stat Phys 120, 1–36 (2005). https://doi.org/10.1007/s10955-005-5534-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-005-5534-3