Skip to main content
SpringerLink
Log in

The energy density in the planar Ising model

  • Published:
Acta Mathematica

Abstract

We study the critical Ising model on the square lattice in bounded simply connected domains with + and free boundary conditions. We relate the energy density of the model to a discrete fermionic correlator and compute its scaling limit by discrete complex analysis methods. As a consequence, we obtain a simple exact formula for the scaling limit of the energy field one-point function in terms of the hyperbolic metric. This confirms the predictions originating in physics, but also provides a higher precision.

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. Assis, M. & McCoy, B. M., The energy density of an Ising half-plane lattice. J. Phys. A, 44 (2011), 095003, 10 pp.

    Google Scholar 

  2. Baxter, R. J., Exactly Solved Models in Statistical Mechanics. Academic Press, London, 1989.

  3. Boutillier C., de Tilière B.: The critical Z-invariant Ising model via dimers: the periodic case. Probab. Theory Related Fields, 147, 379–413 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Boutillier C., de Tilière B.: The critical Z-invariant Ising model via dimers: locality property. Comm. Math. Phys., 301, 473–516 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Burkhardt T., Guim I.: Conformal theory of the two-dimensional Ising model with homogeneous boundary conditions and with disordered boundary fields. Phys. Rev. B, 47, 14306–14311 (1993)

    Article  Google Scholar 

  6. Cardy J.: Conformal invariance and surface critical behavior. Nucl. Phys. B, 240, 514–532 (1984)

    Article  Google Scholar 

  7. Chelkak, D., Hongler, C. & Izyurov, K., Conformal invariance of spin correlations in the planar Ising model. Preprint, 2012. arXiv:1202.2838 [math-ph].

  8. Chelkak D., Smirnov S.: Discrete complex analysis on isoradial graphs. Adv. Math. 228, 1590–1630 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chelkak D., Smirnov S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math., 189, 515–580 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. Di Francesco, P., Mathieu, P. & Sénéchal, D., Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer, New York, 1997.

  11. Grimmett, G., The Random-Cluster Model. Grundlehren der Mathematischen Wissenschaften, 333. Springer, Berlin–Heidelberg, 2006.

  12. Hecht R.: Correlation functions for the two-dimensional Ising model. Phys. Rev., 158, 557–561 (1967)

    Article  Google Scholar 

  13. Hongler, C., Conformal Invariance of Ising Model Correlations. Ph.D. Thesis, Université de Genéve, Genéve, 2010.

  14. Kadanoff, L.P. & Ceva, H., Determination of an operator algebra for the two-dimensional Ising model. Phys. Rev. B, 3 (1971), 3918–3939.

    Google Scholar 

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

    Article  MATH  Google Scholar 

  16. Kenyon R.: Conformal invariance of domino tiling. Ann. Probab., 28, 759–795 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kesten, H., Hitting probabilities of random walks on Z d. Stochastic Process. Appl., 25 (1987), 165–184.

    Google Scholar 

  18. Kramers, H.A. & Wannier, G.H., Statistics of the two-dimensional ferromagnet. I. Phys. Rev., 60 (1941), 252–262.

    Google Scholar 

  19. McCoy, B. M. & Wu, T. T., Theory of Toeplitz determinant and spin correlations of the two-dimensional Ising model IV. Phys. Rev., 162 (1967), 436–475.

    Google Scholar 

  20. McCoy, B. M. & Wu, T. T., The Two-Dimensional Ising Model. Harvard University Press, Cambridge, MA, 1973.

  21. Mercat C.: Discrete Riemann surfaces and the Ising model. Comm. Math. Phys., 218, 177–216 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  23. Palmer, J., Planar Ising Correlations. Progress in Mathematical Physics, 49. Birkhäuser, Boston, MA, 2007.

  24. Smirnov, S., Towards conformal invariance of 2D lattice models, in International Congress of Mathematicians. Vol. II, pp. 1421–1451. Eur. Math. Soc., Zürich, 2006.

  25. Smirnov S.: Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. of Math., 172, 1435–1467 (2010)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Columbia University, 2990 Broadway, New York, NY, 10027, U.S.A.

    Clément Hongler

  2. Section de Mathématiques, Université de Genève, 2–4 rue du Lièvre, Case postale 64, CH-1211, Genève 4, Switzerland

    Stanislav Smirnov

  3. Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, St. Petersburg, 199178, Russia

    Stanislav Smirnov

Authors
  1. Clément Hongler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stanislav Smirnov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Clément Hongler.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hongler, C., Smirnov, S. The energy density in the planar Ising model. Acta Math 211, 191–225 (2013). https://doi.org/10.1007/s11511-013-0102-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11511-013-0102-1

Keywords

Search

Navigation