A functional approach to the numerical conformal bootstrap

TL;DR

This work advocates and demonstrates a functional-basis approach to the numerical 1D conformal bootstrap, showing that bases dual to generalized free solutions capture the correct asymptotics and drastically accelerate convergence relative to the traditional derivative basis. By analyzing extremal solutions and their dual functionals, the authors argue that truncations with a handful of components can reproduce tight bounds on OPE data, including φ^4-flow-like interpolations, with convergence scaling expected as ~1/N^6 when high-dimension tails align with generalized free behavior. They validate the method with two applications: bootstrapping the generalized free boson using a fermionic basis and constructing universal OPE bounds that interpolate between bosonic and fermionic spectra; in both cases, a small number of functionals suffices to achieve high accuracy. The results motivate extending these functionals to higher dimensions, where they promise substantial practical gains by encoding generalized free asymptotics across spins and channels along the z=bar{z} line.

Abstract

We apply recently constructed functional bases to the numerical conformal bootstrap for 1D CFTs. We argue and show that numerical results in this basis converge much faster than the traditional derivative basis. In particular, truncations of the crossing equation with even a handful of components can lead to extremely accurate results, in opposition to hundreds of components in the usual approach. We explain how this is a consequence of the functional basis correctly capturing the asymptotics of bound-saturating extremal solutions to crossing. We discuss how these methods can and should be implemented in higher dimensional applications.

Figures (8)

• Figure 1: Schematic form of the optimal functional. Such a functional provides a valid upper bound on the OPE coefficient at $\Delta_0$ for any choice of $\Delta_g$ down to where the functional first becomes negative.
• Figure 2: Spectrum of the solution saturating an OPE maximization bound for $\Delta_0=2{\Delta_\phi}$, with ${\Delta_\phi}=1/2$, using the fermionic functional basis (top) and derivatives (bottom). As $N$ is increased by two units a new operator appears, while previous operators approach their correct values shown as dashed lines. With functionals the initial error is smaller and convergence is faster.
• Figure 3: Spectrum of the solution saturating OPE maximization bound at $\Delta_0=2{\Delta_\phi}$ with ${\Delta_\phi}=1/2$. $\delta \Delta_n \equiv | \Delta_n- \Delta_n^B|$. Left: using the fermionic functional basis, as $N$ is increased by two units a new operator appears with initial bounded absolute error in dimension, which converges to about $\sim 0.6$ at large $N$. As $N$ increases further this error then rapidly decreases. Right: Convergence of scaling dimension $\Delta_1$. In red the derivative basis, in blue the functional basis. The lines represent fits to a $1/N$ and $1/N^2$ behaviour respectively.
• Figure 4: The single bin functional $\cup$, here shown for ${\Delta_\phi}=\frac{1}{2}$. Contributions to the OPE in the region where the functional is negative must cancel those where it is positive. Given our gap assumptions on the extremal solution, this implies an operator must appear between $2{\Delta_\phi}+1$ and $2{\Delta_\phi}+3$.
• Figure 5: OPE maximization at $\Delta_{\phi}=1$ with $N=21$ components, although no visible changes are seen beyond $N=5$. Main plot: bound on the OPE coefficient of the first operator as a function of its dimension $\Delta_0$. Inset: dimension of the second operator $\Delta_1$ at extremality as a function of $\Delta_0$. In both plots, the dashed lines represent analytic perturbation theory computations up to cubic order in $g\equiv \Delta_1-2{\Delta_\phi}$, as described in the main text.