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

TL;DR

This work determines the critical data of the 3D $\mathcal{N}=1$ minimal SCFT by enforcing emergent supersymmetry within the numerical bootstrap, using four-point functions of the superfield $\Sigma$ and its descendants. By incorporating SUSY relations among OPE coefficients and imposing a spectrum with a single relevant T-parity even scalar and two relevant T-parity odd scalars, the authors carve an isolated island in $(\Delta_{\sigma},\Delta_{\sigma'})$ and extract high-precision exponents: $\Delta_{\sigma}=0.584444(30)$, $\Delta_{\sigma'}=2.882(9)$, giving $η_{\sigma}=η_{\psi}=0.168888(60)$, $1/ν=1.415556(30)$, and $ω=0.882(9)$. They also determine $C_T^{\mathcal{N}=1}/C_T^{f.s.} \approx 1.684$, in line with four-loop $\epsilon$-expansion estimates, and discuss cross-checks with large-$N$ Padé resummations and previous bootstrap bounds. The results support identifying the island with the 3D $\mathcal{N}=1$ minimal model and point to future Monte Carlo and lattice studies as well as extensions to flavor symmetries and higher SUSY theories. This work demonstrates that enforcing SUSY constraints in bootstrap can sharply pin down nontrivial SCFT fixed points with emergent symmetries.

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 $σ$.

Figures (2)

• Figure 1: Bound on the scaling dimension of the subleading T-parity odd operator. We assume that the spectrum contains only one relevant T-parity even operator. The numerics is performed at $\Lambda=13$.
• Figure 2: The region that allowed a unitary ${\cal N}=1$ SCFT to exist. We assume that the spectrum contains only one relevant T-parity even operator, and $\sigma$ and $\sigma'$ are the only two relevant T-parity odd scalar operators. The numerics is performed at $\Lambda=27$.