Symmetry Breaking from Monopole Condensation in QED$_3$

QED in three dimensions with an $SU(2)_f$ doublet $\psi^i$ of massless, charge-1 Dirac fermions (and no Chern-Simons term) has a $U(2) = (SU(2)_f \times U(1)_m)/\mathbb{Z}_2$ symmetry that acts on gauge-invariant local operators, including monopole operators charged under $U(1)_m$. We establish that there are only two possible IR scenarios: either the theory flows to a CFT with $U(2)$ symmetry (a scenario strongly constrained by conformal bootstrap bounds); or it spontaneously breaks $U(2) \to U(1)$ via the condensation of a monopole operator of smallest $U(1)_m$ charge, which is a $U(2)$ doublet. This leads to three Nambu-Goldstone bosons described by a sigma model into a squashed three-sphere $S^3$ with $U(2)$ isometry. We further show that the conventional $SU(2)_f$-triplet order parameter $i \bar \psi \vec \sigma \, \psi$ also gets a vev, exactly aligned with the monopole vev, such that the triplet parametrizes the $\mathbb{CP}^1$ base of the $S^3$ Hopf bundle, with the monopoles providing the $S^1$ fibers. We also recall why this scenario is compatible with the Vafa-Witten theorem. We obtain these results by analyzing the phase diagram as a function of the fermion triplet mass $\vec m$: we show that for all $\vec m \neq 0$ there is a Coulomb phase with only a weakly-coupled photon at low energies, arising from a monopole vev that is aligned with $\vec m$ via the Hopf map. We then argue that taking $\vec m \to 0$ leads to the symmetry-breaking scenario above. Throughout, we give a detailed account of anomaly matching, which leads to a $\theta=\pi$ term in the $S^3$ sigma model. In one presentation, it can be understood as a Hopf term in a suitably gauged version of the $\mathbb{CP}^1$ sigma model.