Truncatable bootstrap equations in algebraic form and critical surface exponents

Abstract

We describe examples of drastic truncations of conformal bootstrap equations encoding much more information than that obtained by a direct numerical approach. A three-term truncation of the four point function of a free scalar in any space dimensions provides algebraic identities among conformal block derivatives which generate the exact spectrum of the infinitely many primary operators contributing to it. In boundary conformal field theories, we point out that the appearance of free parameters in the solutions of bootstrap equations is not an artifact of truncations, rather it reflects a physical property of permeable conformal interfaces which are described by the same equations. Surface transitions correspond to isolated points in the parameter space. We are able to locate them in the case of 3d Ising model, thanks to a useful algebraic form of 3d boundary bootstrap equations. It turns out that the low-lying spectra of the surface operators in the ordinary and the special transitions of 3d Ising model form two different solutions of the same polynomial equation. Their interplay yields an estimate of the surface renormalization group exponent, $y_h=0.72558(18)$ for the ordinary universality class and $y_h=1.646(2)$ for the special universality class, which compare well with the most recent Monte Carlo calculations. Estimates of other surface exponents as well as OPE coefficients are also obtained.

17 pages, 2 figures. v2: typos corrected, references added, acknowledgements updated

References in corpus (12)

• Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
• Bootstrapping Mixed Correlators in the 3D Ising Model
• Bounds on 4D Conformal and Superconformal Field Theories
• The $\mathcal{N}=8$ Superconformal Bootstrap in Three Dimensions
• Central Charge Bounds in 4D Conformal Field Theory
• Critical exponents of the 3d Ising and related models from Conformal Bootstrap
• Bootstrapping phase transitions in QCD and frustrated spin systems
• Five dimensional $O(N)$-symmetric CFTs from conformal bootstrap
• Bootstrapping $O(N)$ Vector Models in $4<d<6$
• The thermodynamic Casimir force: A Monte Carlo study of the crossover between the ordinary and the normal surface universality class
• A Semidefinite Program Solver for the Conformal Bootstrap
• Four loop renormalization of phi^3 theory in six dimensions

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• On co-dimension two defect operators