Accurate boundary bootstrap for the three-dimensional O($N$) normal universality class

TL;DR

This paper advances the boundary conformal bootstrap for the 3d O($N$) normal universality class by employing an $\eta$-minimization approach to the truncated boundary bootstrap. By solving the zeroes of a cost function across high truncation orders $\Lambda$ and incorporating both bulk $S$ and $T$ exchanges, the authors obtain highly accurate BOE coefficients $(a_{\phi}, b_{\phi t})$ and the RG parameter $\alpha$ for $N=2,3,4,5$, in excellent agreement with Monte Carlo data and with new predictions for $N=4,5$. They extend the method to the Ising BCFT ($N=1$), achieving state-of-the-art precision for Ising boundary data by leveraging precise bulk inputs, and demonstrate the method’s robustness for non-positivity-constrained bootstrap problems. The results resolve previous tensions between bootstrap and MC studies, refine the boundary data landscape, and highlight the potential of $\eta$ minimization for a broad class of nonunitary or positivity-violating bootstrap targets.

Abstract

The three-dimensional classical O($N$) model with a boundary has received renewed interest due to the discovery of the extraordinary-log boundary universality class for $2\leq N< N_\text{c}$. The critical value $N_c$ and the exponent of the boundary correlation function are related to certain amplitudes in the normal universality class. To determine their precise values, we revisit the 3d O($N$) boundary conformal field theory for $N=1, 2, 3, 4, 5$. After substantially improving the accuracy of the boundary bootstrap, our determinations are in excellent agreement with the Monte Carlo results, resolving the previous discrepancies due to low truncation orders. We also use the recent bulk bootstrap results to deduce highly accurate Ising data. Many bulk and boundary predictions are obtained for the first time. Our results demonstrate the great potential of the $η$ minimization method for many unexplored bootstrap problems in which positivity constraints are absent.

Figures (8)

• Figure 1: Crossing symmetry of $\langle \mathcal{O}_1\,\mathcal{O}_2\rangle$ in boundary CFT.
• Figure 2: The $N=2,3,4,5$ estimates for the universal amplitudes $(a_{\phi},b_{\phi t})$. Our results (stars) show excellent agreement with the Monte Carlo estimates (dots) for $N=2,3$Toldin:2021kun and resolve the discrepancies seen in the previous Conformal Bootstrap study Padayasi:2021sik (triangles).
• Figure 3: The universal RG parameter $\alpha$ in \ref{['alpha-definition']} at various truncation orders $\Lambda$ (dots). For comparison, we also plot the Monte Carlo results with errors Toldin:2021kun (dashed lines), and the previous Conformal Bootstrap results Padayasi:2021sik (triangles, $\Lambda=9$).
• Figure 4: Power-law fits of some 3d Ising results from the $\langle\sigma\,\epsilon\rangle$ crossing equation. The input-induced errors are indicated by error bars, which are smaller than the plot symbols for $\Delta_{\sigma'}$.
• Figure 5: The 3d Ising 1-point coefficients $(a_{\sigma},a_{\epsilon})$ from this work (purple), Monte Carlo simulations Przetakiewicz:2025gzi (green), and the previous Conformal Bootstrap Gliozzi:2015qsa (orange). The uncertainty in our estimate for $a_\epsilon$ is smaller than the symbol size.