Journal of Statistical Physics

, Volume 157, Issue 4–5, pp 869–914 | Cite as

Solving the 3d Ising Model with the Conformal Bootstrap II. \(c\)-Minimization and Precise Critical Exponents

  • Sheer El-Showk
  • Miguel F. Paulos
  • David Poland
  • Slava RychkovEmail author
  • David Simmons-Duffin
  • Alessandro Vichi


We use the conformal bootstrap to perform a precision study of the operator spectrum of the critical 3d Ising model. We conjecture that the 3d Ising spectrum minimizes the central charge \(c\) in the space of unitary solutions to crossing symmetry. Because extremal solutions to crossing symmetry are uniquely determined, we are able to precisely reconstruct the first several \(\mathbb {Z}_2\)-even operator dimensions and their OPE coefficients. We observe that a sharp transition in the operator spectrum occurs at the 3d Ising dimension \(\Delta _\sigma = 0.518154(15)\), and find strong numerical evidence that operators decouple from the spectrum as one approaches the 3d Ising point. We compare this behavior to the analogous situation in 2d, where the disappearance of operators can be understood in terms of degenerate Virasoro representations.


Critical phenomena Conformal invariance Ising Model Critical exponents Central charge Stress tensor 



We are grateful to M. Hasenbusch, M. Henkel, D. Mouhanna and E. Vicari for the useful communications concerning their work. We are grateful to B. van Rees for the discussions of the interpolating solution. In addition, we thank N. Arkani-Hamed, C. Beem, A. L. Fitzpatrick, G. Fleming, H. Ooguri, H. Osborn, J. Kaplan, E. Katz, F. Kos, J. Maldacena, J. Penedones, L. Rastelli, N. Seiberg, and A. Zhiboedov for related discussions. S. R. is grateful to the Samara Chernorechenskaya Scientific Center for their hospitality. S. R., D. S. D., and D. P. are grateful to KITP for their hospitality. We would also like to thank the organizers and participants of the Back to the Bootstrap 3 conference at CERN. This research was supported in part by the National Science Foundation under Grant No. PHY11-25915. The work of S. E. was partially supported by the French ANR contract 05-BLAN-NT09-573739, the ERC Advanced Grant no. 226371 and the ITN programme PITN-GA-2009-237920. M.P. is supported by DOE Grant DE-FG02-11ER41742. A. V is supported by DOE Grant DE-AC02-05CH1123. The work of D. S. D. is supported by DOE Grant number DE-SC0009988. Computations for this paper were run on National Energy Research Scientific Computing Center supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH1123; on the CERN cluster; on the Aurora and Hyperion clusters supported by the School of Natural Sciences Computing Staff at the Institute for Advanced Study; on the Omega cluster supported by the facilities and staff of the Yale University Faculty of Arts and Sciences High Performance Computing Center; on the TED cluster of the Chemistry Department and High Energy Theory group at Brown University; and the Kelvin cluster at the C. E. A. Saclay funded by the European Research Council Advanced Investigator Grant ERCAdG228301. S. E. would like to thank D. Kosower for providing access to the Kelvin cluster.


  1. 1.
    El-Showk, S., Paulos, M.F., Poland, D., Rychkov, S., Simmons-Duffin, D., Vichi, A.: Solving the 3D ising model with the conformal bootstrap. Phys. Rev. D86, 025022 (2012). arXiv:1203.6064 [hep-th]
  2. 2.
    El-Showk, S., Paulos, M.F.: Bootstrapping conformal field theories with the extremal functional method. Phys. Rev. Lett. 111, 241601 (2012). arXiv:1211.2810 [hep-th]CrossRefADSGoogle Scholar
  3. 3.
    El-Showk, S., Paulos, M.F., Poland, D., Rychkov, S., Simmons-Duffin, D., Vichi, A.: Conformal field theories in fractional dimensions. Phys. Rev. Lett. 112, 141601 (2014). arXiv:1309.5089 [hep-th]
  4. 4.
    Kos, F., Poland, D., Simmons-Duffin, D.: Bootstrapping the \(O(N)\) vector models. arXiv:1307.6856 [hep-th]
  5. 5.
    Kos, F., Poland, D., Simmons-Duffin, D.: Bootstrapping Mixed Correlators in the 3D Ising Model. arXiv:1406.4858 [hep-th]
  6. 6.
    Heemskerk, I., Penedones, J., Polchinski, J., Sully, J.: Holography from conformal field theory. JHEP 0910, 079 (2009). arXiv:0907.0151[hep-th]MathSciNetCrossRefADSGoogle Scholar
  7. 7.
    Fitzpatrick, A.L., Kaplan, J.: Unitarity and the holographic S-Matrix. JHEP 1210, 032 (2012). arXiv:1112.4845 [hep-th]CrossRefADSGoogle Scholar
  8. 8.
    Rattazzi, R., Rychkov, V.S., Tonni, E., Vichi, A.: Bounding scalar operator dimensions in 4D CFT. JHEP 12, 031 (2008). arXiv:0807.0004 [hep-th]MathSciNetCrossRefADSGoogle Scholar
  9. 9.
    Pappadopulo, D., Rychkov, S., Espin, J., Rattazzi, R.: OPE convergence in conformal field theory. Phys. Rev. D86, 105043 (2012). arXiv:1208.6449 [hep-th]ADSGoogle Scholar
  10. 10.
    Poland, D., Simmons-Duffin, D.: Bounds on 4D conformal and superconformal field theories. JHEP 1105, 017 (2011). arXiv:1009.2087 [hep-th]MathSciNetCrossRefADSGoogle Scholar
  11. 11.
    Rattazzi, R., Rychkov, S., Vichi, A.: Central charge bounds in 4D conformal field theory. Phys. Rev. D83, 046011 (2011). arXiv:1009.2725 [hep-th]ADSGoogle Scholar
  12. 12.
    Poland, D., Simmons-Duffin, D., Vichi, A.: Carving out the space of 4D CFTs. JHEP 1205, 110 (2012). arXiv:1109.5176 [hep-th]CrossRefADSGoogle Scholar
  13. 13.
    Nishioka, T., Yonekura, K.: On RG flow of \(\tau _{RR}\) for supersymmetric field theories in three-dimensions. JHEP 1305, 165 (2013). arXiv:1303.1522 [hep-th]MathSciNetCrossRefADSGoogle Scholar
  14. 14.
    Rychkov, S.: Conformal bootstrap in three dimensions? arXiv:1111.2115 [hep-th]
  15. 15.
    Maldacena, J., Zhiboedov, A.: Constraining conformal field theories with a higher spin symmetry. J. Phys. A46, 214011 (2013). arXiv:1112.1016 [hep-th]MathSciNetADSGoogle Scholar
  16. 16.
    Campostrini, M., Pelissetto, A., Rossi, P., Vicari, E.: Improved high-temperature expansion and critical equation of state of three-dimensional ising-like systems. Phys. Rev. E 60, 3526–3563 (1999). arXiv:cond-mat/9905078 [cond-mat]
  17. 17.
    Wilson, K., Kogut, J.B.: The Renormalization group and the epsilon expansion. Phys. Rept. 12, 75–200 (1974)CrossRefADSGoogle Scholar
  18. 18.
    Fitzpatrick, A.L., Kaplan, J., Poland, D., Simmons-Duffin, D.: The analytic bootstrap and AdS superhorizon locality. JHEP 1312, 004 (2013). arXiv:1212.3616 [hep-th]MathSciNetCrossRefADSGoogle Scholar
  19. 19.
    Komargodski, Z., Zhiboedov, A.: Convexity and liberation atlLarge spin. JHEP 1311, 140 (2013). arXiv:1212.4103 [hep-th]CrossRefADSGoogle Scholar
  20. 20.
    Callan, J., Curtis, G., Gross, D.J.: Bjorken scaling in quantum field theory. Phys. Rev. D8, 4383–4394 (1973)Google Scholar
  21. 21.
    Nachtmann, O.: Positivity constraints for anomalous dimensions. Nucl. Phys. B63, 237–247 (1973)CrossRefADSGoogle Scholar
  22. 22.
    Guida, R., Zinn-Justin, J.: Critical exponents of the \(N\) vector model. J. Phys. A 31, 8103–8121 (1998). arXiv:cond-mat/9803240 [cond-mat]
  23. 23.
    Campostrini, M., Pelissetto, A., Rossi, P., Vicari, E.: 25th-order high-temperature expansion results for three-dimensional ising-like systems on the simple-cubic lattice. Phys. Rev. E 65, 066127 (2002). arXiv:cond-mat/0201180 [cond-mat]CrossRefADSGoogle Scholar
  24. 24.
    Deng, Y., Blöte, H.W.J.: Simultaneous analysis of several models in the three-dimensional ising universality class. Phys. Rev. E 68, 036125 (2003)CrossRefADSGoogle Scholar
  25. 25.
    Hasenbusch, M.: Finite size scaling study of lattice models in the three-dimensional ising universality class. Phys. Rev. B 82, 174433 (2010). arXiv:1004.4486 [cond-mat]CrossRefADSGoogle Scholar
  26. 26.
    Canet, L., Delamotte, B., Mouhanna, D., Vidal, J.: Nonperturbative renormalization group approach to the Ising model: a derivative expansion at order \(\partial ^4\). Phys. Rev. B68, 064421 (2003). arXiv:hep-th/0302227 [hep-th]CrossRefADSGoogle Scholar
  27. 27.
    Litim, D.F., Zappala, D.: Ising exponents from the functional renormalisation group. Phys. Rev. D83, 085009 (2011). arXiv:1009.1948 [hep-th]ADSGoogle Scholar
  28. 28.
    Newman, K.E., Riedel, E.K.: Critical exponents by the scaling-field method: the isotropic \(O(N)\)-vector model in three dimensions. Phys. Rev. B 30, 6615–6638 (1984)CrossRefADSGoogle Scholar
  29. 29.
    Comellas, J., Travesset, A.: O (N) models within the local potential approximation. Nucl. Phys. B498, 539–564 (1997). arXiv:hep-th/9701028[hep-th]
  30. 30.
    Litim, D.F.: Critical exponents from optimized renormalization group flows. Nucl. Phys. B 631, 128–158 (2002). arXiv:hep-th/0203006 [hep-th].
  31. 31.
    Litim, D.F., Vergara, L.: Subleading critical exponents from the renormalization group. Phys. Lett. B 581, 263–269 (2004). arXiv:hep-th/0310101 [hep-th]
  32. 32.
    Henkel, M.: Finite size scaling and universality in the (2+1)-dimensions ising model. J. Phys. A20, 3969 (1987)ADSGoogle Scholar
  33. 33.
    Brower, R., Fleming, G., Neuberger, H.: Lattice radial quantization: 3D Ising. Phys. Lett. B721, 299–305 (2013). arXiv:1212.6190 [hep-lat]MathSciNetCrossRefADSGoogle Scholar
  34. 34.
    Weigel, M., Janke, W.: Universal amplitude ratios in finite-size scaling: three-dimensional ising model. Nucl. Phys. B (Proc. Suppl.) 83, 721 (2000). arXiv:cond-mat/0009032 CrossRefADSGoogle Scholar
  35. 35.
    Hathrell, S.J.: Trace anomalies and \(\lambda \phi ^4\) theory in curved space. Ann. Phys. 139, 136 (1982)MathSciNetCrossRefADSGoogle Scholar
  36. 36.
    Jack, I., Osborn, H.: Background field calculations in curved space-time. 1. Generalf formalism and application to scalar fields. Nucl. Phys. B234, 331 (1984)CrossRefADSGoogle Scholar
  37. 37.
    Cappelli, A., Friedan, D., Latorre, J.I.: C theorem and spectral representation. Nucl. Phys. B352, 616–670 (1991)MathSciNetCrossRefADSGoogle Scholar
  38. 38.
    Petkou, A.: Conserved currents, consistency relations, and operator product expansions in the conformally invariant \(O(N)\) vector model. Ann. Phys. 249, 180–221 (1996). arXiv:hep-th/9410093 MathSciNetCrossRefzbMATHADSGoogle Scholar
  39. 39.
    Le Guillou, J., Zinn-Justin, J.: Accurate critical exponents for Ising like systems in noninteger dimensions. J. Phys. 48, 19–24 (1987)CrossRefGoogle Scholar
  40. 40.
    Codello, A.: Scaling solutions in continuous dimension. J. Phys. A45, 465006 (2012). arXiv:1204.3877 [hep-th]MathSciNetADSGoogle Scholar
  41. 41.
    Rychkov, V.S., Vichi, A.: Universal constraints on conformal operator dimensions. Phys. Rev. D80, 045006 (2009). arXiv:0905.2211 [hep-th]MathSciNetADSGoogle Scholar
  42. 42.
    Vichi, A.: A new method to explore conformal field theories in any dimension. Ph.D. Thesis, EPFL, 2011, 164 pp. url:
  43. 43.
    Liendo, P., Rastelli, L., van Rees, B.C.: The bootstrap program for boundary CFT\({}_d\). JHEP 1307, 113 (2013). arXiv:1210.4258 [hep-th]CrossRefADSGoogle Scholar
  44. 44.
    Friedan, D., Qiu, Z.-A., Shenker, S.H.: Conformal invariance, unitarity and two-dimensional critical exponents. Phys. Rev. Lett. 52, 1575–1578 (1984)MathSciNetCrossRefADSGoogle Scholar
  45. 45.
    Friedan, D., Shenker, S.H., Qiu, Z.-A.: Details of the nonunitarity proof for highest weight representations of the Virasoro Algebra. Commun. Math. Phys. 107, 535 (1986)MathSciNetCrossRefzbMATHADSGoogle Scholar
  46. 46.
    Hogervorst, M., Rychkov, S.: Radial coordinates for conformal blocks. Phys. Rev. D87, 106004 (2013). arXiv:1303.1111 [hep-th]ADSGoogle Scholar
  47. 47.
    Hogervorst, M., Osborn, H., Rychkov, S.: Diagonal limit for conformal blocks in \(d\) dimensions. JHEP 1308, 014 (2013). arXiv:1305.1321 [hep-th]MathSciNetCrossRefADSGoogle Scholar
  48. 48.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes: The Art of Scientific Computing. Univ. Pr, Cambridge (2007)Google Scholar
  49. 49.
    Reemtsen, R., Görner, S.: Numerical methods for semi-infinite programming: a survey. In: Reemtsen, R., Rückmann, J.-J. (eds.) Semi-Infinite Programming, pp. 195–275. Springer, Berlin (1998)Google Scholar
  50. 50.
  51. 51.
  52. 52.
    Vidal, G.: Entanglement renormalization: an introduction. arXiv:0912.1651 [cond-mat.str-el]
  53. 53.
    Vicari, E.: Critical phenomena and renormalization-group flow of multi-parameter \(\Phi ^4\) field theories. PoS LAT2007 (2007) 023. arXiv:0709.1014 [hep-lat]
  54. 54.
  55. 55.
    Rattazzi, R., Rychkov, S., Vichi, A.: Bounds in 4D conformal field theories with global symmetry. J. Phys. A44, 035402 (2011). arXiv:1009.5985[hep-th]MathSciNetADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Sheer El-Showk
    • 1
  • Miguel F. Paulos
    • 2
  • David Poland
    • 3
  • Slava Rychkov
    • 1
    • 4
    Email author
  • David Simmons-Duffin
    • 5
  • Alessandro Vichi
    • 6
  1. 1.Theory DivisionCERNGenevaSwitzerland
  2. 2.Department of PhysicsBrown UniversityProvidenceUSA
  3. 3.Department of PhysicsYale UniversityNew HavenUSA
  4. 4.Faculté de PhysiqueUniversité Pierre et Marie Curie & Laboratoire de Physique Théorique, École Normale SupérieureParisFrance
  5. 5.School of Natural SciencesInstitute for Advanced StudyPrincetonUSA
  6. 6.Theoretical Physics Group, Ernest Orlando Lawrence Berkeley National Laboratory and Center for Theoretical PhysicsUniversity of California, BerkeleyBerkeleyUSA

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