Precision Islands in the Ising and $O(N)$ Models

TL;DR

The paper advances the conformal bootstrap program by enforcing uniqueness of leading operators in 3d CFTs through a scan over the ratio of leading OPE coefficients, producing much tighter islands for the Ising and $O(N)$ vector models. Using mixed correlators and semidefinite programming, they obtain high-precision values for the leading scaling dimensions and OPE coefficients, notably $Δ_σ=0.5181489(10)$, $Δ_ε=1.412625(10)$, and $λ_{εεε}/λ_{σσε}=1.456889(50)$ for the 3d Ising model, and precise results for $O(2)$ and $O(3)$ models as well. The approach yields competitive or superior precision compared with Monte Carlo or some experimental inputs, and reveals tensions with certain measurements in the $O(2)$ case, highlighting the bootstrap’s potential to sharpen our understanding of critical phenomena and guide future studies of CFT spectra. The method also provides OPE-coefficient determinations as byproducts and suggests broader applicability to other theories and conformal windows.

Abstract

We make precise determinations of the leading scaling dimensions and operator product expansion (OPE) coefficients in the 3d Ising, $O(2)$, and $O(3)$ models from the conformal bootstrap with mixed correlators. We improve on previous studies by scanning over possible relative values of the leading OPE coefficients, which incorporates the physical information that there is only a single operator at a given scaling dimension. The scaling dimensions and OPE coefficients obtained for the 3d Ising model, $(Δ_σ, Δ_ε,λ_{σσε}, λ_{εεε}) = (0.5181489(10), 1.412625(10), 1.0518537(41), 1.532435(19))$, give the most precise determinations of these quantities to date.

Figures (7)

• Figure 1: Determination of the leading scaling dimensions in the 3d Ising model from the mixed correlator bootstrap after scanning over the ratio of OPE coefficients $\lambda_{\epsilon\epsilon\epsilon}/\lambda_{\sigma\sigma\epsilon}$ and projecting to the $(\Delta_{\sigma}, \Delta_{\epsilon})$ plane (blue region). Here we assume that $\sigma$ and $\epsilon$ are the only relevant $\mathbb{Z}_2$-odd and $\mathbb{Z}_2$-even scalars, respectively. In this plot we compare to the previous best Monte Carlo determinations Hasenbusch:2011yya (dashed rectangle). This region is computed at $\Lambda=43$.
• Figure 2: Allowed islands from the mixed correlator bootstrap for the $O(2)$ and $O(3)$ models after scanning over the ratio of OPE coefficients $\lambda_{sss}/\lambda_{\phi\phi s}$ and projecting to the ($\Delta_{\phi},\Delta_{s}$) plane (blue regions). Here we assume that $\phi$ and $s$ are the only relevant scalar operators in their $O(N)$ representations. These islands are computed at $\Lambda=35$. The Ising island is marked with a cross because it is too small to see on the plot.
• Figure 3: Determination of the leading scaling dimensions $(\Delta_{\sigma},\Delta_{\epsilon})$ and the OPE coefficient ratio $\lambda_{\epsilon\epsilon\epsilon}/\lambda_{\sigma\sigma\epsilon}$ in the 3d Ising model from the mixed correlator bootstrap (blue region). This region is computed at $\Lambda=43$.
• Figure 4: Determination of the leading OPE coefficients in the 3d Ising model from the conformal bootstrap (blue region). This region was obtained by computing upper and lower bounds on the OPE coefficient magnitude at $\Lambda=27$, for points in the allowed region of figure \ref{['fig:3dIsingIsland']}.
• Figure 5: Allowed islands from the mixed correlator bootstrap for $N=2$ after scanning over the OPE coefficient ratio $\lambda_{sss}/\lambda_{\phi\phi s}$ and projecting to the $(\Delta_{\phi}, \Delta_s)$ plane (blue regions). Here we assumed that $\phi$ and $s$ are the only relevant operators in their $O(N)$ representations. These islands are computed at $\Lambda=19,27,35$. The green rectangle shows the Monte Carlo determination from Campostrini:2006ms, while the horizontal lines show the $1\sigma$ (solid) and $3\sigma$ (dashed) confidence intervals from experiment Lipa:2003zz.