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Role of quenched randomness in metallic quantum criticality is one of thelong standing problems in condensed matter physics. An aspect of thefundamental difficulties lies in the fact that such nonmagnetic disorders leadeffective interactions between abundant soft modes near the Fermi surface to bedrastically enhanced particularly in the diffusive regime, where theperturbative framework does not work. Here, we revisit the problem of dirtyquantum critical metals in a different angle, focusing on the hydrodynamicregime instead of the diffusive regime near the non-Fermi liquid quantumcritical point. More concretely, we study effects of mutual correlationsbetween quantum critical nematic fluctuations and weak localizationcorrections, and show the existence of a weak-disorder non-Fermi liquid fixedpoint, based on the renormalization group (RG) analysis up to the two-looporder. The two-loop order RG analysis suggests that the absence of quantumcoherence in two-particle composite excitations weakens the role of weaklocalization corrections and allows a weakly disordered non-Fermi liquidmetallic state in the hydrodynamic regime of the nematic quantum criticalpoint. Although this dirty non-Fermi liquid metallic state may not be the trueinfrared stable fixed point at zero temperature, expected to be characterizedby the diffusive Ohmic regime, we argue that this weak-disorder non-Fermiliquid metallic fixed point would govern the region of an intermediate energyscale, described by effective hydrodynamics of nematic quantum criticality. Inthis respect we believe that this research would be an important step inunderstanding the $T-$linear electrical resistivity as a characteristic featureof non-Fermi liquids and the origin of unconventional superconductivity fromeffective hydrodynamics of quantum criticality.

Existence of a weak-disorder non-Fermi liquid fixed point in the hydrodynamic regime of two-dimensional nematic quantum criticality