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 Rychkov
  • David Simmons-Duffin
  • Alessandro Vichi
Article

Abstract

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.

Keywords

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

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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
  • 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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