Acta Mathematica

, Volume 211, Issue 2, pp 191–225 | Cite as

The energy density in the planar Ising model

Article

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.

Keywords

Ising model energy density discrete analytic function fermions conformal invariance hyperbolic geometry conformal field theory 

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References

  1. AM.
    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. B.
    Baxter, R. J., Exactly Solved Models in Statistical Mechanics. Academic Press, London, 1989.Google Scholar
  3. BT1.
    Boutillier C., de Tilière B.: The critical Z-invariant Ising model via dimers: the periodic case. Probab. Theory Related Fields, 147, 379–413 (2010)CrossRefMATHMathSciNetGoogle Scholar
  4. BT2.
    Boutillier C., de Tilière B.: The critical Z-invariant Ising model via dimers: locality property. Comm. Math. Phys., 301, 473–516 (2011)CrossRefMATHMathSciNetGoogle Scholar
  5. BG.
    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)CrossRefGoogle Scholar
  6. C.
    Cardy J.: Conformal invariance and surface critical behavior. Nucl. Phys. B, 240, 514–532 (1984)CrossRefGoogle Scholar
  7. CHI.
    Chelkak, D., Hongler, C. & Izyurov, K., Conformal invariance of spin correlations in the planar Ising model. Preprint, 2012. arXiv:1202.2838 [math-ph].
  8. CS1.
    Chelkak D., Smirnov S.: Discrete complex analysis on isoradial graphs. Adv. Math. 228, 1590–1630 (2011)CrossRefMATHMathSciNetGoogle Scholar
  9. CS2.
    Chelkak D., Smirnov S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math., 189, 515–580 (2012)CrossRefMATHMathSciNetGoogle Scholar
  10. DMS.
    Di Francesco, P., Mathieu, P. & Sénéchal, D., Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer, New York, 1997.Google Scholar
  11. G.
    Grimmett, G., The Random-Cluster Model. Grundlehren der Mathematischen Wissenschaften, 333. Springer, Berlin–Heidelberg, 2006.Google Scholar
  12. He.
    Hecht R.: Correlation functions for the two-dimensional Ising model. Phys. Rev., 158, 557–561 (1967)CrossRefGoogle Scholar
  13. Ho.
    Hongler, C., Conformal Invariance of Ising Model Correlations. Ph.D. Thesis, Université de Genéve, Genéve, 2010.Google Scholar
  14. KC.
    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. Ka.
    Kaufman B.: Crystal statistics. II. Partition function evaluated by spinor analysis. Phys. Rev., 76, 1232–1243 (1949)CrossRefMATHGoogle Scholar
  16. Ken.
    Kenyon R.: Conformal invariance of domino tiling. Ann. Probab., 28, 759–795 (2000)CrossRefMATHMathSciNetGoogle Scholar
  17. Kes.
    Kesten, H., Hitting probabilities of random walks on Z d. Stochastic Process. Appl., 25 (1987), 165–184.Google Scholar
  18. KW.
    Kramers, H.A. & Wannier, G.H., Statistics of the two-dimensional ferromagnet. I. Phys. Rev., 60 (1941), 252–262.Google Scholar
  19. MW1.
    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. MW2.
    McCoy, B. M. & Wu, T. T., The Two-Dimensional Ising Model. Harvard University Press, Cambridge, MA, 1973.Google Scholar
  21. M.
    Mercat C.: Discrete Riemann surfaces and the Ising model. Comm. Math. Phys., 218, 177–216 (2001)CrossRefMATHMathSciNetGoogle Scholar
  22. O.
    Onsager, L., Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev., 65 (1944), 117–149.Google Scholar
  23. P.
    Palmer, J., Planar Ising Correlations. Progress in Mathematical Physics, 49. Birkhäuser, Boston, MA, 2007.Google Scholar
  24. S1.
    Smirnov, S., Towards conformal invariance of 2D lattice models, in International Congress of Mathematicians. Vol. II, pp. 1421–1451. Eur. Math. Soc., Zürich, 2006.Google Scholar
  25. S2.
    Smirnov S.: Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. of Math., 172, 1435–1467 (2010)MATHMathSciNetGoogle Scholar

Copyright information

© Institut Mittag-Leffler 2013

Authors and Affiliations

  1. 1.Department of MathematicsColumbia UniversityNew YorkU.S.A.
  2. 2.Section de MathématiquesUniversité de GenèveGenève 4Switzerland
  3. 3.Chebyshev LaboratorySt. Petersburg State UniversitySt. PetersburgRussia

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