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 $η_σ=0.168888(60)$ and $ω=0.882(9)$. 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 $σ$ is highly restricted. If we further assume the SCFT to have only two time-reversal parity odd relevant operators, $σ$ and $σ'$, we find that allowed region for $Δ_σ$ and $Δ_{σ'}$ becomes an isolated island. The result is obtained by considering not only the four point correlator $\langle σσσσ\rangle$, but also $\langle σεσε\rangle$ and $\langle εεεε\rangle$, with $ε\sim σ^2$ being the superconformal descendant of $σ$.