Communications in Mathematical Physics

, Volume 322, Issue 2, pp 303–332 | Cite as

Holomorphic Spinor Observables in the Critical Ising Model

Article

Abstract

We introduce a new version of discrete holomorphic observables for the critical planar Ising model. These observables are holomorphic spinors defined on double covers of the original multiply connected domain. We compute their scaling limits, and show their relation to the ratios of spin correlations, thus providing a rigorous proof to a number of formulae for those ratios predicted by CFT arguments.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. BDC12.
    Beffara, V., Duminil-Copin, H.: Smirnov’s fermionic observable away from criticality. Ann. Prob. 40(6), 2667–2689 (2012)Google Scholar
  2. BG93.
    Burkhardt T., Guim I.: Conformal theory of the two-dimensional Ising model with homogeneous boundary conditions and with disordred boundary fields. Phys. Rev. B 47(21), 14306–14311 (1993)ADSCrossRefGoogle Scholar
  3. CHI12.
    Chelkak, D., Hongler, C., Izyurov, K.: Conformal invariance of spin correlations in the planar Ising model. http://arxiv.org/abs/1202.2838v1 [math-ph], 2012
  4. CS11.
    Chelkak D., Smirnov S.: Discrete complex analysis on isoradial graphs. Adv. in Math. 228(3), 1590–1630 (2011)MathSciNetMATHCrossRefGoogle Scholar
  5. CS12.
    Chelkak D., Smirnov S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Inv. Math. 189(3), 515–580 (2012)MathSciNetADSMATHCrossRefGoogle Scholar
  6. DHN11.
    Duminil-Copin H., Hongler C., Nolin P.: Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model. Comm. Pure Appl. Math. 64(9), 1165–1198 (2011)MathSciNetMATHCrossRefGoogle Scholar
  7. Hon10.
    Hongler, C.: Conformal invariance of Ising model correlations. Ph.D. thesis, 2010, available at www.math.columbia.edu/~hongler/thesis.pdf
  8. HP12.
    Hongler C., Phong D.H.: Hardy spaces and boundary conditions from the Ising model. Math. Zeit. 274, 209–224 (2013)MathSciNetMATHCrossRefGoogle Scholar
  9. HS10.
    Hongler, C., Smirnov, S.: The energy density in the planar Ising model. http://arxiv.org/abs/1008.2645v2 [math-ph], 2010
  10. Izy11.
    Izyurov, K.: Holomorphic spinor observables and interfaces in the critical Ising model Ph.D. thesis, 2011Google Scholar
  11. KC71.
    Kadanoff, L.P., Ceva, H.L.: Determination of an operator algebra for the two-dimensional Ising model. Phys. Rev. B (3) 3, 3918–3939 (1971)Google Scholar
  12. Ken11a.
    Kenyon R.: Spanning forests and the vector bundle laplacian. Ann. Prob. 39(5), 1983–2017 (2011)MathSciNetMATHCrossRefGoogle Scholar
  13. Ken11b.
    Kenyon, R.: Conformal invariance of loops in the double-dimer model. http://arxiv.org/abs/1105.4158v2 [math.PR], 2012
  14. Mer01.
    Mercat C.: Discrete Riemann surfaces and the Ising model. Commun. Math. Phys. 218(1), 177–216 (2001)MathSciNetADSMATHCrossRefGoogle Scholar
  15. Pal07.
    Palmer, J.: Planar Ising correlations, Volume 49 of Progress in Mathematical Physics. Boston, MA: Birkhäuser Boston Inc., 2007Google Scholar
  16. Pom92.
    Pommerenke, C.: Boundary behaviour of conformal maps, Berlin-Heidelberg-NewYork: Springer-Ferlag, 1992Google Scholar
  17. RC06.
    Riva, V., Cardy, J.: Holomorphic parafermions in the Potts model and stochastic Loewner evolution. J. Stat. Mech. Theory Exp., 12, P12001, 19 pp. (electronic) (2006)Google Scholar
  18. Smi06.
    Smirnov, S.: Towards conformal invariance of 2D lattice models. In: International Congress of Mathematicians. Vol. II, Zürich: Eur. Math. Soc., 2006, pp. 1421–1451Google Scholar
  19. Smi10.
    Smirnov, S.: Conformal invariance in random cluster models. I. Holmorphic fermions in the Ising model. Ann. Math. (2) 172, 1435–1467 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.St.Petersburg Department of Steklov Mathematical Institute (PDMI RAS)St.PetersburgRussia
  2. 2.Section de Mathématiques, Université de GenèveGenève 4Suisse
  3. 3.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland
  4. 4.Chebyshev Laboratory, Department of Mathematics and MechanicsSaint-Petersburg State UniversitySaint-PetersburgRussia

Personalised recommendations