On conformality and self-duality of $N_f=2$ QED$_3$

We study the IR phase of three dimensional quantum electrodynamics (QED$_3$) coupled to $N_f=2$ flavors of two-component Dirac fermions, which has been controversial for decades. This theory has been proposed to be self-dual with symmetry enhancement $(SU(2)_f\times U(1)_t )/\mathbb{Z}_2\rightarrow O(4)$ at the IR fixed point. We focus on the four-point correlator of monopole operators with unit topological charge of $U(1)_t$. We show that this four-point correlator indeed can be embedded in an $O(4)$ symmetric form and the $O(4)\rightarrow SU(2)_f\times U(1)_t $ branching rules are well consistent with the self-duality. We use conformal bootstrap method to derive dynamical constraints on the CFT data and test the conformality and self-duality of $N_f=2$ QED$_3$. In particular we find the CFT data obtained from previous lattice simulations can be ruled out by introducing irrelevant assumptions in the spectrum, indicating the IR phase of $N_f=2$ QED$_3$ is not conformal.