Bootstrapping the Minimal 3D SCFT

Abstract

We study the conformal bootstrap constraints for 3D conformal field theories with a $\mathbb{Z}_2$ or parity symmetry, assuming a single relevant scalar operator $ε$ that is invariant under the symmetry. When there is additionally a single relevant odd scalar $σ$, we map out the allowed space of dimensions and three-point couplings of such "Ising-like" CFTs. If we allow a second relevant odd scalar $σ'$, we identify a feature in the allowed space compatible with 3D $\mathcal{N}=1$ superconformal symmetry and conjecture that it corresponds to the minimal $\mathcal{N}=1$ supersymmetric extension of the Ising CFT. This model has appeared in previous numerical bootstrap studies, as well as in proposals for emergent supersymmetry on the boundaries of topological phases of matter. Adding further constraints from 3D $\mathcal{N}=1$ superconformal symmetry, we isolate this theory and use the numerical bootstrap to compute the leading scaling dimensions $Δ_σ = Δ_ε - 1 = .58444(22)$ and three-point couplings $λ_{σσε} = 1.0721(2)$ and $λ_{εεε} = 1.67(1)$. We additionally place bounds on the central charge and use the extremal functional method to estimate the dimensions of the next several operators in the spectrum. Based on our results we observe the possible exact relation $λ_{εεε}/λ_{σσε} = \tan(1)$.

Figures (6)

• Figure 1: A plot of the allowed region of 3D CFTs with a $\mathbb{Z}_2$ symmetry. Here we assume two relevant operators, $\sigma$ and $\epsilon$, taken to be $\mathbb Z_2$-odd and $\mathbb Z_2$-even symmetric, respectively. This plot assumes the permutation symmetry $\lambda_{\sigma\epsilon\sigma} = \lambda_{\sigma\sigma\epsilon}$. The shaded area is not excluded. In this plot we use $\Lambda=25$.
• Figure 2: A plot showing how the allowed range of $\theta$, taken over all $\Delta_\epsilon$, grows on the non-excluded region of CFTs. In this plot we use $\Lambda=25$.
• Figure 3: The shaded areas show the non-excluded regions at various gap assumptions. Here we assume a third relevant parity-odd operator $\sigma'$, together with the relevant operators $\sigma$ and $\epsilon$ from before. We additionally assume that there are no other parity-even scalar operators below $\Delta_{\epsilon'} = \Delta_{\sigma'} + 1$. In this plot we use $\Lambda=25$.
• Figure 4: (Left) A plot of the lower bound on the central charge $C_T$ versus $\Delta_\sigma$, constrained to the SUSY line $\Delta_\sigma = \Delta_\epsilon - 1$ at gaps $\Delta_{\sigma'} = \Delta_{\epsilon'} -1 \geq \{2.75, 2.80, 2.85, 2.875 \}$. (Right) The lower bound of $C_T$ along the additional constraint of Equation \ref{['eq:susy']}. In these plots we use $\Lambda=25$.
• Figure 5: A plot of the non-excluded regions with gap assumptions of $\Delta_{\sigma'} = \Delta_{\epsilon'}-1 \geq 2.8$ along the supersymmetric line $\Delta_\epsilon = \Delta_\sigma + 1$. The shaded regions are obtained at derivative order $\Lambda = \{9, 13, 17, 21, 25\}$ and denote the non-excluded points.