Symmetry Breaking from Monopole Condensation in QED${}_3$

Abstract

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 argue that there are only two plausible 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. 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 recall why this scenario is compatible with the Vafa-Witten theorem. These results are obtained by considering the phase diagram as a function of the fermion triplet mass $\vec{m}$. We argue that for all $\vec{m}\ne 0$ there is a Coulomb phase with a weakly-coupled photon, which arises from a suitable monopole vev; 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.

65 pages; clarifications added, typos fixed

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