  We study the effect of disorder on massless, spinful Dirac fermions in two
spatial dimensions with attractive interactions, and show that the combination
of disorder and attractive interactions is deadly to the Dirac semimetal phase.
First, we derive the zero temperature phase diagram of a clean Dirac fermion
system with tunable doping level ({\mu}) and attraction strength (g). We show
that it contains two phases: a superconductor and a Dirac semimetal. Then, we
show that arbitrarily weak disorder destroys the Dirac semimetal, turning it
into a superconductor. We discuss the strength of the superconductivity for
both long range and short range disorder. For long range disorder, the
superconductivity is exponentially weak in the disorder strength. For short
range disorder, a uniform mean field analysis predicts that superconductivity
should be doubly exponentially weak in the disorder strength. However, a more
careful treatment of mesoscopic fluctuations suggests that locally
superconducting puddles should form at a much higher temperature, and should
establish global phase coherence at a temperature that is only exponentially
small in weak disorder. We also discuss the effect of disorder on the quantum
critical point of the clean system, building in the effect of disorder through
a replica field theory. We show that disorder is a relevant perturbation to the
supersymmetric quantum critical point. We expect that in the presence of
attractive interactions, the flow away from the critical point ends up in the
superconducting phase, although firm conclusions cannot be drawn since the
renormalization group analysis flows to strong coupling. We argue that although
we expect the quantum critical point to get buried under a superconducting
phase, signatures of the critical point may be visible in the finite
temperature quantum critical regime.
