  Role of quenched randomness in metallic quantum criticality is one of the
long standing problems in condensed matter physics. An aspect of the
fundamental difficulties lies in the fact that such nonmagnetic disorders lead
effective interactions between abundant soft modes near the Fermi surface to be
drastically enhanced particularly in the diffusive regime, where the
perturbative framework does not work. Here, we revisit the problem of dirty
quantum critical metals in a different angle, focusing on the hydrodynamic
regime instead of the diffusive regime near the non-Fermi liquid quantum
critical point. More concretely, we study effects of mutual correlations
between quantum critical nematic fluctuations and weak localization
corrections, and show the existence of a weak-disorder non-Fermi liquid fixed
point, based on the renormalization group (RG) analysis up to the two-loop
order. The two-loop order RG analysis suggests that the absence of quantum
coherence in two-particle composite excitations weakens the role of weak
localization corrections and allows a weakly disordered non-Fermi liquid
metallic state in the hydrodynamic regime of the nematic quantum critical
point. Although this dirty non-Fermi liquid metallic state may not be the true
infrared stable fixed point at zero temperature, expected to be characterized
by the diffusive Ohmic regime, we argue that this weak-disorder non-Fermi
liquid metallic fixed point would govern the region of an intermediate energy
scale, described by effective hydrodynamics of nematic quantum criticality. In
this respect we believe that this research would be an important step in
understanding the $T-$linear electrical resistivity as a characteristic feature
of non-Fermi liquids and the origin of unconventional superconductivity from
effective hydrodynamics of quantum criticality.
