Skip to main content
SpringerLink
Log in

Algebraic Reduction of the Ising Model

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We consider the Ising model on a cylindrical lattice of L columns, with fixed-spin boundary conditions on the top and bottom rows. The spontaneous magnetization can be written in terms of partition functions on this lattice. We show how we can use the Clifford algebra of Kaufman to write these partition functions in terms of L by L determinants, and then further reduce them to m by m determinants, where m is approximately L/2. In this form the results can be compared with those of the Ising case of the superintegrable chiral Potts model. They point to a way of calculating the spontaneous magnetization of that more general model algebraically.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Onsager, L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65, 117–149 (1944)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. Kaufman, B.: Crystal statistics. II. Partition function evaluated by spinor analysis. Phys. Rev. 76, 1232–1243 (1949)

    Article  MATH  ADS  Google Scholar 

  3. Onsager, L.: In: Proceedings of the IUPAP Conference on Statistical Mechanics, “Discussione e observazioni”. Nuovo Cimento (Suppl.), Ser. 9 6, 261 (1949)

  4. Montroll, E.W., Potts, R.B., Ward, J.C.: Correlations and spontaneous magnetization of the two-dimensional Ising model. J. Math. Phys. 4, 308–322 (1963)

    Article  MathSciNet  ADS  Google Scholar 

  5. Yang, C.N.: The spontaneous magnetization of a two-dimensional Ising model. Phys. Rev. 85, 808–816 (1952)

    Article  MATH  ADS  Google Scholar 

  6. Baxter, R.J.: Derivation of the order parameter of the chiral Potts model. Phys. Rev. Lett. 94, 130602 (2005)

    Article  ADS  Google Scholar 

  7. Baxter, R.J.: The order parameter of the chiral Potts model. J. Stat. Phys. 120, 1–36 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. Grenander, U., Szegő, G.: Toeplitz Forms and Their Applications. Univ. Calif. Press, Berkeley (1958)

    MATH  Google Scholar 

  9. Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic, London (1989). Dover Publications, Mineola (2007)

    MATH  Google Scholar 

  10. Baxter, R.J.: Corner transfer matrices of the eight-vertex model. II. The Ising model case. J. Stat. Phys. 17, 1–14 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  11. Baxter, R.J.: Superintegrable chiral Potts model: Thermodynamic properties, an “inverse” model, and a simple associated hamiltonian. J. Stat. Phys. 57, 1–39 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  12. Baxter, R.J.: A conjecture for the superintegrable chiral Potts model. J. Stat. Phys. 132 (2008). doi:10.1007/s10955-008-9588-x

Download references

Author information

Authors and Affiliations

  1. Mathematical Sciences Institute, The Australian National University, Canberra, ACT, 0200, Australia

    R. J. Baxter

Authors
  1. R. J. Baxter
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Baxter, R.J. Algebraic Reduction of the Ising Model. J Stat Phys 132, 959–982 (2008). https://doi.org/10.1007/s10955-008-9587-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-008-9587-y

Keywords

Search

Navigation