Bootstrapping the Minimal 3D SCFT

TL;DR

This paper uses the conformal bootstrap to identify and analyze the minimal 3D $ ext{N}=1$ supersymmetric extension of the Ising model, providing precise operator dimensions and couplings.

Contribution

It maps out the allowed parameter space for 3D Ising-like CFTs with supersymmetry and isolates the minimal $ ext{N}=1$ supersymmetric model using bootstrap constraints.

Findings

Computed the leading scaling dimensions: $ au_ ext{sigma} = 0.58444(22)$

Determined three-point couplings: $ ext{lambda}_{ ext{sigma} ext{sigma} ext{epsilon}}$ and $ ext{lambda}_{ ext{epsilon} ext{epsilon} ext{epsilon}}$

Estimated operator spectrum and bounds on the central charge.

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 $\epsilon$ that is invariant under the symmetry. When there is additionally a single relevant odd scalar $\sigma$, 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 $\sigma'$, 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 $\Delta_{\sigma} = \Delta_{\epsilon} - 1 = .58444(22)$ and three-point couplings $\lambda_{\sigma\sigma\epsilon} = 1.0721(2)$ and $\lambda_{\epsilon\epsilon\epsilon} = 1.67(1)$. 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 $\lambda_{\epsilon\epsilon\epsilon}/\lambda_{\sigma\sigma\epsilon} = \tan(1)$.