Advertisement

Communications in Mathematical Physics

, Volume 367, Issue 3, pp 771–833 | Cite as

Ising Model: Local Spin Correlations and Conformal Invariance

  • Reza GheissariEmail author
  • Clément Hongler
  • S. C. Park
Open Access
Article
  • 65 Downloads

Abstract

We study the 2-dimensional Ising model at critical temperature on a simply connected subset \({\Omega_{\delta}}\) of the square grid \({\delta\mathbb{Z}^{2}}\). The scaling limit of the critical Ising model is conjectured to be described by Conformal Field Theory; in particular, there is expected to be a precise correspondence between local lattice fields of the Ising model and the local fields of Conformal Field Theory. Towards the proof of this correspondence, we analyze arbitrary spin pattern probabilities (probabilities of finite spin configurations occurring at the origin), explicitly obtain their infinite-volume limits, and prove their conformal covariance at the first (non-trivial) order. We formulate these probabilities in terms of discrete fermionic observables, enabling the study of their scaling limits. This generalizes results of Hongler (Conformal invariance of Ising model correlations. Ph.D. thesis, [Hon10]), Hongler and Smirnov (Acta Math 211(2):191–225, [HoSm13]), Chelkak, Hongler, and Izyurov (Ann. Math. 181(3), 1087–1138, [CHI15]) to one-point functions of any local spin correlations. We introduce a collection of tools which allow one to exactly and explicitly translate any spin pattern probability (and hence any lattice local field correlation) in terms of discrete complex analysis quantities. The proof requires working with multipoint lattice spinors with monodromy (including construction of explicit formulae in the full plane), and refined analysis near their source points to prove convergence to the appropriate continuous conformally covariant functions.

Notes

Acknowledgements

Firstly, the authors thank the anonymous referees for their many helpful suggestions and comments. This research was initiated during the Research Experience for Undergraduates program at Mathematics Department of Columbia University, funded by the NSF under grant DMS-0739392. We would like to thank the T.A., Krzysztof Putyra, for his help during this program and the program coordinator Robert Lipshitz, as well as all the participants, in particular Adrien Brochard and Woo Chang Chung. R.G. would like to thank Chuck Newman and Eyal Lubetzky for interesting discussions. C.H. would like to thank Dmitry Chelkak and Stanislav Smirnov, for sharing many ideas and insights about the Ising model and conformal invariance; Itai Benjamini and Curtis McMullen for asking questions that suggested we look at this problem; Stéphane Benoist, John Cardy, Julien Dubédat, Hugo Duminil-Copin, Konstantin Izyurov, Kalle Kytölä and Wendelin Werner for interesting discussions, the NSF under grant DMS-1106588, the ERC under grant SG CONSTAMIS, the NCCR Swissmap, the Blavatnik family foundation, the Latsis family foundation, and the Minerva Foundation for financial support. S.P. would like to thank C.M. Munteanu for interesting discussions.

References

  1. Bax89.
    Baxter, R. Exactly Solved Models in Statistical Mechanics Harcourt Brace Jovanovich Publishers, Academic Press Inc. London 1989Google Scholar
  2. BPZ84.
    Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in twodimensional quantum field theory. Nucl. Phys. B 241(2), 333–380 (1984)Google Scholar
  3. Bo07.
    Boutillier C.: Pattern densities in non-frozen dimer models. Commun. Math. Phys. 271(1), 55–91 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. BeHo16.
    Benoist, S., Hongler, C.: The scaling limit of critical Ising interfaces is CLE(3) (to appear in Ann. Probab.) (2016)Google Scholar
  5. BeHo18.
    Benoist, S., Hongler, C.: in preparationGoogle Scholar
  6. BoDT09.
    Boutillier C., de Tilière B.: The critical Z-invariant Ising model via dimers: locality property. Commun. Math. Phys. 301(2), 473–516 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. BoDT08.
    Boutillier C., de Tilière B.: The critical Z-invariant Ising model via dimers: the periodic case. Prob. Theory Relat. Fields 147(3), 379–413 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. BuGu93.
    Burkhardt T., Guim I.: Conformal theory of the two-dimensional Ising model with homogeneous boundary conditions and with disordered boundary fields. Phys. Rev. B (1) 47, 14306–14311 (1993)ADSCrossRefGoogle Scholar
  9. Car84.
    Cardy J.: Conformal invariance and surface critical behavior. Nucl. Phys. B 240, 514–532 (1984)ADSCrossRefGoogle Scholar
  10. CDHKS14.
    Chelkak D., Duminil-Copin H., Hongler C., Kemppainen A., Smirnov S.: Convergence of Ising Interfaces to Schramm’s SLE curves. C. R. Math. Acad. Sci. Paris 352(2), 157–161 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. CCK17.
    Chelkak, D., Cimasoni, D., Kassel, A.: Revisiting the combinatorics of the 2D Ising model. Annales de l’Institut Henri Poincaré (D) 4(3), 309–385 (2017)Google Scholar
  12. CGS17.
    Chelkak, D., Glazman, A., Smirnov, S.: Discrete stress-energy tensor in the loop O(n) model (2017). arXiv:1604.06339
  13. ChHo18.
    Chelkak, D., Hongler, C.: in preparationGoogle Scholar
  14. CHI15.
    Chelkak D., Hongler C., Izyurov K.: Conformal invariance of spin correlations in the planar Ising model. Ann. Math. 181(3), 1087–1138 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. ChIz13.
    Chelkak D., Izyurov K.: Holomorphic spinor observables in the critical Ising model. Commun. Math. Phys. 322(2), 302–303 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. ChSm11.
    Chelkak D., Smirnov S.: Discrete complex analysis on isoradial graphs. Adv. Math. 228, 1590–1630 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. ChSm12.
    Chelkak D., Smirnov S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189(3), 515–580 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. CFL28.
    Courant R., Friedrichs K., Lewy, H.: Uber die partiellen Differenzengleichungen der mathematischen. Phys. Math. Ann. 100, 32–74 (1928)Google Scholar
  19. DMS97.
    Di Francesco, P., Mathieu, P., Sénéchal, D.: Conformal Field Theory, Graduate Texts in Contemporary Physics. Springer, New York (1997)Google Scholar
  20. DHN11.
    Duminil-Copin H., Hongler C., Nolin P.: Connection probabilities and RSW-type bounds for the FK Ising model. Commun. Pure Appl. Math. 64(9), 1165–1198 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Dub15.
    Dubédat J.: Dimers and families of Cauchy–Riemann operators I. J. Am. Math. Soc. 28, 1063–1167 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Gri06.
    Grimmett, G.: The Random-Cluster Model. Volume 333 of Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (2006)Google Scholar
  23. Hon10.
    Hongler, C.: Conformal Invariance of Ising model Correlations. Ph.D. thesis (2010)Google Scholar
  24. HoKy10.
    Hongler, C., Kytölä, K.: Ising Interfaces and Free Boundary Conditions. J. Am. Math. Soc. 26(4), 1107–1189 (2013).  https://doi.org/10.1090/S0894-0347-2013-00774-2
  25. HKV17.
    Hongler, C., Kytölä, K., Viklund, F.: Conformal Field Theory at the Lattice Level: Discrete Complex Analysis and Virasoro Structure, arXiv:1307.4104 (2017)
  26. HoSm13.
    Hongler C., Smirnov S.: The energy density in the critical planar Ising model. Acta Math. 211(2), 191–225 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Isi25.
    Ising E.: Beitrag zur Theorie des Ferromagnetismus. Z. Phys. 31, 253–258 (1925)ADSCrossRefGoogle Scholar
  28. KaCe71.
    Kadanoff L., Ceva H.: Determination of an operator algebra for the two-dimensional Ising model. Phys. Rev. B (3) 3, 3918–3939 (1971)ADSMathSciNetCrossRefGoogle Scholar
  29. Kau49.
    Kaufman B.: Crystal statistics. II. Partition function evaluated by spinor analysis. Phys. Rev., II. Ser. 76, 1232–1243 (1949)ADSzbMATHGoogle Scholar
  30. KaOn49.
    Kaufman B., Onsager L.: Crystal statistics. III. Short-range order in a binary Ising lattice. Phys. Rev. II. Ser. 76, 1244–1252 (1949)ADSzbMATHGoogle Scholar
  31. KaOn50.
    Kaufman, B., Onsager, L.: Crystal statistics. IV. Long-range order in a binary crystal. Unpublished typescript (1950)Google Scholar
  32. Ken00.
    Kenyon R.: Conformal invariance of domino tiling. Ann. Probab. 28, 759–795 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  33. Kes87.
    Kesten H.: Hitting probabilities of random walks on \({\mathbb{Z}^{d}}\). Stoch. Processes Appl. 25, 165–184 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  34. KrWa41.
    Kramers H.A., Wannnier G.H.: Statistics of the two-dimensional ferromagnet. I.. Phys. Rev. (2) 60, 252–262 (1941)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. LaLi04.
    Lawler G.F., Limic V.: The Beurling estimate for a class of random walks. Electron. J. Probab. 9, 846–861 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  36. Lel55.
    Lelong-Ferrand, J.: Représentation conforme et transformations à intègrale de Dirichlet bornée. Gauthier-Villars, Paris (1955)Google Scholar
  37. Len20.
    Lenz W.: Beitrag zum Verständnis der magnetischen Eigenschaften in festen Körpern. Phys. Z. 21, 613–615 (1920)Google Scholar
  38. LuSl12.
    Lubetzky E., Sly A.: Critical Ising on the square lattice mixes in polynomial time. Commun. Math. Phys. 313, 815–836 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. Mer01.
    Mercat C.: Discrete Riemann surfaces and the Ising model. Commun. Math. Phys. 218, 177–216 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. McWu73.
    McCoy B.M., Wu T.T.: The Two-Dimensional Ising Model. Harvard University Press, Cambridge, MA (1973)CrossRefzbMATHGoogle Scholar
  41. Ons44.
    Onsager L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. (2) 65, 117–149 (1944)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. Pal07.
    Palmer J.: Planar Ising correlations. Birkhäuser, Basel (2007)zbMATHGoogle Scholar
  43. SMJ77.
    Sato, M., Miwa, T., Jimbo, M.: Studies on holonomic quantum fields, I-IV. Proc. Jpn Acad. Ser. A Math. Sci. 53(1), 6–10; 53(1), 147–152; 53(1), 153–158; 53(1), 183–185 (1977)Google Scholar
  44. Smi06.
    Smirnov, S.: Towards conformal invariance of 2D lattice models. In: Sanz-Solé, M. et al.(ed.) Proceedings of the International congress of Mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume II: Invited lectures, pp. 1421–1451. Zürich: European Mathematical Society (EMS) (2006)Google Scholar
  45. Smi10.
    Smirnov S.: Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. Math. 172(2), 1435–1467 (2007)MathSciNetzbMATHGoogle Scholar
  46. Yan52.
    Yang C.N.: The spontaneous magnetization of a two-dimensional Ising model. Phys. Rev. (2) 85, 808–816 (1952)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019
Corrected Publication March/2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Ecole Polytechnique Fédérale de LausanneEPFL SB MATHAA CSFTLausanneSwitzerland

Personalised recommendations