Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-23T06:51:43.153Z Has data issue: false hasContentIssue false
  • English
  • Français

PROOF OF THE DETERMINANTAL FORM OF THE SPONTANEOUS MAGNETIZATION OF THE SUPERINTEGRABLE CHIRAL POTTS MODEL

Part of: Equilibrium statistical mechanics

Published online by Cambridge University Press:  25 November 2010

R. J. BAXTER
R. J. BAXTER*
Affiliation:
Mathematical Sciences Institute, The Australian National University, Canberra, A.C.T. 0200, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The superintegrable chiral Potts model has many resemblances to the Ising model, so it is natural to look for algebraic properties similar to those found for the Ising model by Onsager, Kaufman and Yang. The spontaneous magnetization ℳr can be written in terms of a sum over the elements of a matrix Sr. The author conjectured the form of the elements, and this conjecture has been verified by Iorgov et al. The author also conjectured in 2008 that this sum could be expressed as a determinant, and has recently evaluated the determinant to obtain the known result for ℳr. Here we prove that the sum and the determinant are indeed identical expressions. Since the order parameters of the superintegrable chiral Potts model are also those of the more general solvable chiral Potts model, this completes the algebraic calculation of ℳr for the general model.

Keywords

statistical mechanicslattice modelsspontaneous magnetizationdeterminant

MSC classification

Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
Type
Research Article
Information
The ANZIAM Journal , Volume 51 , Issue 3 , January 2010 , pp. 309 - 316
Copyright
Copyright © Australian Mathematical Society 2010

References

[1]Albertini, G., McCoy, B. M., Perk, J. H. H. and Tang, S., “Excitation spectrum and order parameter for the integrable N-state chiral Potts model”, Nuclear Phys. B 314 (1989) 741–763.Google Scholar
[2]Au-Yang, H. and Perk, J. H. H., “Quantum loop subalgebra and eigenvectors of the superintegrable chiral Potts transfer matrices”, 2009, arXiv0907.0362.Google Scholar
[3]Au-Yang, H. and Perk, J. H. H., “Spontaneous magnetization of the integrable chiral Potts model”, 2010, arXiv1003.4805.Google Scholar
[4]Au-Yang, H. and Perk, J. H. H., “Identities in the superintegrable chiral Potts model”, J. Phys. A: Math. Theor. 43 (2010) 025203 (10pp).Google Scholar
[5]Baxter, R. J., “Free energy of the solvable chiral Potts model”, J. Stat. Phys. 52 (1988) 639–667.CrossRefGoogle Scholar
[6]Baxter, R. J., “Chiral Potts model: eigenvalues of the transfer matrix”, Phys. Lett. A 146 (1990) 110–114.CrossRefGoogle Scholar
[7]Baxter, R. J., “Derivation of the order parameter of the chiral Potts model”, Phys. Rev. Lett. 94 (2005) 130602 (3pp).CrossRefGoogle ScholarPubMed
[8]Baxter, R. J., “The order parameter of the chiral Potts model”, J. Stat. Phys. 120 (2005) 1–36.Google Scholar
[9]Baxter, R. J., “Algebraic reduction of the Ising model”, J. Stat. Phys. 132 (2008) 959–982.CrossRefGoogle Scholar
[10]Baxter, R. J., “A conjecture for the superintegrable chiral Potts model”, J. Stat. Phys. 132 (2008) 983–1000.CrossRefGoogle Scholar
[11]Baxter, R. J., “Some remarks on a generalization of the superintegrable chiral Potts model”, J. Stat. Phys. 137 (2009) 798–813.CrossRefGoogle Scholar
[12]Baxter, R. J., “Spontaneous magnetization of the superintegrable chiral Potts model: calculation of the determinant D PQ”, J. Phys. A: Math. Theor. 43 (2010) 145002 (16pp).CrossRefGoogle Scholar
[13]Hurst, C. A. and Green, H. S., “New solution of the Ising problem for a rectangular lattice”, J. Chem. Phys. 33 (1960) 1059–1062.Google Scholar
[14]Iorgov, N., Shadura, V., Tykhyy, Yu., Pakuliak, S. and von Gehlen, G., “Spin operator matrix elements in the superintegrable chiral Potts quantum chain”, J. Stat. Phys. 139 (2010) 743–768.Google Scholar
[15]Kac, M. and Ward, J. C., “A combinatorial solution of the the two-dimensional Ising model”, Phys. Rev. 88 (1952) 1332–1337.CrossRefGoogle Scholar
[16]Kaufman, B., “Crystal statistics. II. Partition function evaluated by spinor analysis”, Phys. Rev. 76 (1949) 1232–1243.CrossRefGoogle Scholar
[17]Montroll, E. W., Potts, R. B. and Ward, J. C., “Correlations and spontaneous magnetization of the two-dimensional Ising model”, J. Math. Phys. 4 (1963) 308–322.Google Scholar
[18]Onsager, L., “Crystal statistics. I. A two-dimensional model with an order-disorder transition”, Phys. Rev. 65 (1944) 117–149.CrossRefGoogle Scholar
[19]Onsager, L., “‘Discussione e observazioni’, Proceedings of the IUPAP conference on statistical mechanics”, Nuovo Cimento, Suppl, IX Ser. 6 (1949) 261.Google Scholar
[20]Yang, C. N., “The spontaneous magnetization of a two-dimensional Ising model”, Phys. Rev. 85 (1952) 808–816.Google Scholar