Extremal bootstrapping: go with the flow

TL;DR

This work develops extremal-flow bootstrap, a framework where solutions saturating crossing symmetry bounds are characterized by extremality equations and can be deformed along parameter-space boundaries via linearized flow equations. By combining error-correction with these flows, the authors achieve drastic computational speedups and high-precision spectra, even enabling non-unitary bootstrap in $D=1$. They demonstrate upgrading across truncation levels, and continuous flows in external dimensions, achieving remarkable accuracy with modest computing resources. The approach unifies OPE and gap maximization in a single formalism, relates to, and extends, the determinant method, and provides a path to robust, scalable multi-correlator bootstrap. The results imply that extremal theories saturating bounds have sparse spectra and offer practical pathways for exploring non-unitary CFTs and critical phenomena like the Ising model.

Abstract

The extremal functional method determines approximate solutions to the constraints of crossing symmetry, which saturate bounds on the space of unitary CFTs. We show that such solutions are characterized by extremality conditions, which may be used to flow continuously along the boundaries of parameter space. Along the flow there is generically no further need for optimization, which dramatically reduces computational requirements, bringing calculations from the realm of computing clusters to laptops. Conceptually, extremality sheds light on possible ways to bootstrap without positivity, extending the method to non-unitary theories, and implies that theories saturating bounds, and especially those sitting at kinks, have unusually sparse spectra. We discuss several applications, including the first high-precision bootstrap of a non-unitary CFT.

Figures (12)

• Figure 1: Feasible vs unfeasible constraints. In black, the line of vectors labeled by the conformal dimension $\Delta$. The convex hull $H$ of these vectors includes the target $T$ on the left hand side. On the right it does not, and one can find a linear functional (the blue line) which separates the target from the remaining vectors.
• Figure 2: On the left, the extremal case. The functional overlaps with a face of the convex hull of the vectors. On the right, we show how varying $\mathbf T$ smoothly, the extremal functional varies continuously, keeping tangent to $H$.
• Figure 3: Upgrading. Plots show the evolution of the spectrum as the number of crossing constraints $N$ is increased. At the top the conformal dimensions, and on the bottom the corresponding OPE coefficients. For each $N$, cutting the curves with a vertical line gives the spectrum at that $N$. For clarity a few chosen curves are highlighted in color. As $N$ increases new operators appear. Their dimension and OPE coefficient vary a lot in the beginning, but eventually stabilize. The diagonal dashed lines interpolate the successively largest dimension operators, and their OPE coefficients, as a function of $N$. The insets show the leading operator. Notice in particular the leading OPE coefficient converges very fast. Finally, for any $N$ the value $\Delta_{\hbox{\tiny gap}}$ is a valid upper bound, which explains it's decrease with $N$.
• Figure 4: Comparison between extrapolated vs exact spectra. The exact solution is the generalized free fermion with dimensions $\Delta(j)=1+2\Delta_\phi+2j$ (cf. (\ref{['eq:gff']})), here evaluated at $\Delta_\phi=0.3$. The exact values lie on the solid red line whereas the extrapolated results are represented by the blue dots.
• Figure 5: Gap maximization with 100 components. The curve provides a valid upper bound on the dimension of $\phi^2$ in $D=1$ CFTs. The slope of the bound smoothly interpolates between $2$ and $2\sqrt{2}$. As the number of components increases, the transition region is pushed to higher values of $\Delta_\phi$.