Rare region effects dominate weakly disordered three-dimensional Dirac points

We study three-dimensional (3D) Dirac fermions with weak finite-range scalar potential disorder. In the clean system, the density of states vanishes quadratically at the Dirac point. Disorder is known to be perturbatively irrelevant, and previous theoretical work has assumed that the Dirac semimetal phase, characterized by a vanishing density of states, survives at weak disorder, with a finite disorder phase transition to a diffusive metal with a nonvanishing density of states. In this paper, we show that nonperturbative effects from rare regions, which are missed by conventional disorder-averaged calculations, instead give rise to a nonzero density of states for any nonzero disorder. Thus, there is no Dirac semimetal phase at nonzero disorder. The results are established both by a heuristic scaling argument and via a systematic saddle-point analysis. We also discuss transport near the Dirac point. At the Dirac point, we argue that transport is diffusive, and proceeds via hopping between rare resonances. As one moves in chemical potential away from the Dirac point, there are interesting intermediate-energy regimes where the rare regions produce scattering resonances that determine the dc conductivity. We derive a scaling theory of transport near disordered 3D Dirac points. We also discuss the interplay of disorder with attractive interactions at the Dirac point and the resulting granular superconducting and Bose glass phases. Our results are relevant for all 3D systems with Dirac points, including Weyl semimetals.