Truncatable bootstrap equations in algebraic form and critical surface exponents

TL;DR

This work shows that drastic truncations of conformal bootstrap equations can capture far more information than conventional numerical methods, including exact spectra in certain cases, and reveals that boundary CFTs with permeable interfaces naturally exhibit free parameters in truncations. By recasting boundary bootstrap constraints into polynomial equations in $D=3$, the authors derive an algebraic framework where a single quartic in the surface dimension yields the ordinary-transition exponent, and a shared polynomial underlies both ordinary and special Ising transitions. Exploiting the special algebraic form of boundary blocks, they identify distinct solutions corresponding to ordinary and special surface transitions and extract precise surface exponents and OPE data, with results in good agreement with Monte Carlo studies. The approach provides a transparent, analytic handle on surface critical behavior and highlights the role of boundary conditions in shaping the spectrum of surface operators. This method also clarifies the limitations of truncations and suggests a path to refine surface critical data in higher-dimensional or more complex CFTs.

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.

Figures (2)

• Figure 1: Setting $D=3$ and $\Delta_4=\Delta_{\phi^2}+2$ in (\ref{['freetruncation']}) we get a two-dimensional section of the zeros of the 10 $3\times3$ minors made with the first 5 derivatives specified in the text. The common intersection selects the exact solution of the free field theory.
• Figure 2: The scalar product ${\sf v}\cdot f$ of a null left eigenvector of the $8\times8$ minor ${\sf f}$ associated with the truncation (4,4,0) as a function of the scaling dimension of the surface operators.$f$ is the column vector defined in eq.(\ref{['column']}). Notice that ${\sf v}\cdot { f}$ is a polynomial of $8^{\rm th}$ degree in $\Delta$, so it has more zeros than the four values that form a solution of (4,4,0). The additional zeros may be employed to form other solutions.