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

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 $\sigma$.