The conformal bootstrap

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Nature Physics
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The conformal bootstrap was proposed in the 1970s as a strategy for calculating the properties of second-order phase transitions. After spectacular success elucidating two-dimensional systems, little progress was made on systems in higher dimensions until a recent renaissance beginning in 2008. We report on some of the main results and ideas from this renaissance, focusing on new determinations of critical exponents and correlation functions in the three-dimensional Ising and O(N) models.

At a glance


  1. Sum rule geometry.
    Figure 1: Sum rule geometry.
  2. Ising [Delta][epsi] upper bound.
    Figure 2: Ising Δε upper bound.

    Upper bound on the leading -even scalar dimension Δε as a function of the leading -odd scalar dimension Δσ in a 3D CFT21. Although the only assumptions entering this bound are conformal invariance and unitarity, the resulting curve has an obvious kink near the critical dimensions of the 3D Ising model.

  3. Ising critical exponents.
    Figure 3: Ising critical exponents.

    World record determination of the leading scaling dimensions in the 3D Ising model from the conformal bootstrap (blue region) compared to the previous best Monte Carlo determinations (dashed rectangle)26.

  4. O(N) singlet upper bounds.
    Figure 4: O(N) singlet upper bounds.

    Upper bounds on the leading O(N) singlet dimension ΔS as a function of the leading O(N) vector dimension Δϕ, for different values of N. The black pluses show the large N computations (interpolated by the dashed line). The error bars on the curves for the Ising and O(N) models with N = 2,3,4,5,6 show previous determinations using Monte Carlo techniques, the ε-expansion, and high-temperature expansion. See ref. 22 and references therein.

  5. O(N) archipelago.
    Figure 5: O(N) archipelago.

    Allowed islands from the mixed correlator bootstrap for different values of N, assuming the O(N) vector ϕ and singlet S are the only relevant operators in their O(N) representations26. The Ising island is marked with a cross because it is too small to see on the plot.


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  1. Department of Physics, Yale University, New Haven, Connecticut 06520, USA

    • David Poland
  2. School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540, USA

    • David Poland &
    • David Simmons-Duffin

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