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We study three-dimensional Dirac fermions with weak finite-range scalarpotential disorder. We show that even though disorder is perturbativelyirrelevant at 3D Dirac points, nonperturbative effects from rare regions giverise to a nonzero density of states and a finite mean free path, with thetransport at the Dirac point being dominated by hopping between rare regions.As one moves in chemical potential away from the Dirac point, there areinteresting intermediate-energy regimes where the rare regions producescattering resonances that determine the DC conductivity. We also discuss theinterplay of disorder with interactions at the Dirac point. Attractiveinteractions drive a transition into a granular superconductor, with a criticaltemperature that depends strongly on the disorder distribution. In the presenceof Coulomb repulsion and weak retarded attraction, the system can be a Boseglass. Our results apply to all 3D systems with Dirac points, including Weylsemimetals, and overturn a thirty year old consensus regarding the irrelevanceof weak disorder at 3D Dirac points.

Dirty Weyl fermions: rare region effects near 3D Dirac points