Conformal bootstrap bounds for the $U(1)$ Dirac spin liquid and $N=7$ Stiefel liquid

We apply the conformal bootstrap technique to study the $U(1)$ Dirac spin liquid (i.e. $N_f=4$ QED$_3$) and the newly proposed $N=7$ Stiefel liquid (i.e. a conjectured 3d non-Lagrangian CFT without supersymmetry). For the $N_f=4$ QED$_3$, we focus on the monopole operator and ($SU(4)$ adjoint) fermion bilinear operator. We bootstrap their single correlators as well as the mixed correlators between them. We first discuss the bootstrap kinks from single correlators. Some exponents of these bootstrap kinks are close to the expected values of QED$_3$, but we provide clear evidence that they should not be identified as the QED$_3$. We then provide rigorous numerical bounds for the Dirac spin liquid and the $N=7$ Stiefel liquid to be stable critical phases on the triangular and kagome lattice. For the triangular and kagome Dirac spin liquid, the rigorous lower bounds of the monopole operators scaling dimension are $1.046$ and $1.105$, respectively. These bounds are consistent with the latest Monte Carlo results.