  We consider the effects of weak quenched fermionic disorder on the
quantum-phase transition between the Dirac semimetal and charge density wave
(CDW) insulator in two spatial dimensions. The symmetry breaking transition is
described by the Gross-Neveu-Yukawa (GNY) theory of Dirac fermions coupled to
an Ising order parameter field. Treating the disorder using the replica method,
we consider chemical potential, vector potential (gauge), and random mass
disorders, which all arise from non-magnetic charge impurities. We
self-consistently account for the Landau damping of long-wavelength
order-parameter fluctuations by using the non-perturbative RPA re-summation of
fermion loops, and compute the renormalization-group (RG) flow to leading order
in the disorder strength and $1/N$ ($N$ the number of Dirac fermion flavors).
We find two fixed points, the clean GNY critical point which is stable against
weak disorder, and a dirty GNY multi-critical point, at which the chemical
potential disorder is finite and the other forms of disorder are irrelevant. We
investigate the scaling of physical observables at this finite-disorder
multi-critical point which breaks Lorentz invariance and gives rise to distinct
non-Fermi liquid behavior.
