Bootstrapping minimal $\mathcal{N}=1$ superconformal field theory in three dimensions

TL;DR

This paper uses numerical bootstrap methods to precisely determine critical exponents of the minimal 3D $ =1$ superconformal field theory, revealing insights into emergent supersymmetry at a quantum critical point.

Contribution

The work uniquely applies numerical bootstrap with emergent supersymmetry constraints to isolate the SCFT parameters and identify an allowed parameter island.

Findings

Critical exponents $\eta_{\sigma}$ and $\omega$ are precisely determined.

Emergent supersymmetry restricts the conformal dimensions of operators.

An isolated allowed region for operator dimensions is identified under specific assumptions.

Abstract

Using numerical bootstrap method, we determine the critical exponents of the minimal three-dimensional $\mathcal{N}=1$ superconformal field theory (SCFT) to be $\eta_{\sigma}=0.168888(60)$ and $\omega=0.882(9)$. The model was argued in arXiv:1301.7449 to describe a quantum critical point (QCP) at the boundary a $3+1$D topological superconductor. More interestingly, the QCP can be reached by tuning a single parameter, where supersymmetry (SUSY) is realised as an emergent symmetry. By imposing emergent SUSY in numerical bootstrap, we find that the conformal scaling dimension of the real scalar operator $\sigma$ is highly restricted. If we further assume the SCFT to have only two time-reversal parity odd relevant operators, $\sigma$ and $\sigma'$, we find that allowed region for $\Delta_{\sigma}$ and $\Delta_{\sigma'}$ becomes an isolated island. The result is obtained by considering not only the four point correlator $\langle \sigma \sigma \sigma \sigma \rangle$, but also $\langle \sigma \epsilon \sigma \epsilon \rangle$ and $\langle \epsilon \epsilon\epsilon \epsilon \rangle$, with $\epsilon \sim \sigma ^2$ being the superconformal descendant of $\sigma$.