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.
Electrical transport properties near an electronic Ising-nematic quantum critical point in two dimensions are of both theoretical and experimental interest. In this work, we derive a kinetic equation valid in a broad regime near the quantum critical point using the memory matrix approach. The formalism is applied to study the effect of the critical fluctuations on the dc resistivity through different scattering mechanisms, including umklapp, impurity scattering, and electron-hole scattering in a compensated metal. We show that electrical transport in the quantum critical regime exhibits a rich behavior that depends sensitively on the scattering mechanism and the band structure. In the case of a single large Fermi surface, the resistivity due to umklapp scattering crosses over from $\ensuremath{\rho}\ensuremath{\sim}{T}^{2}$ at low temperature to sublinear at high temperature. The crossover temperature scales as
q03, where
q0 is the minimal wave vector for umklapp scattering. Impurity scattering leads to $\ensuremath{\rho}\ensuremath{-}{\ensuremath{\rho}}_{0}\ensuremath{\sim}{T}^{\ensuremath{\alpha}}$ (${\ensuremath{\rho}}_{0}$ being the residual resistivity), where $\ensuremath{\alpha}$ is either larger than 2 if there is only a single Fermi sheet present, or
4/3 in the case of multiple Fermi sheets. Finally, in a perfectly compensated metal with an equal density of electrons and holes, the low-temperature behavior depends strongly on the structure of ``cold spots'' on the Fermi surface, where the coupling between the quasiparticles and order parameter fluctuations vanishes by symmetry. In particular, for a system where cold spots are present on some (but not all) Fermi sheets, $\ensuremath{\rho}\ensuremath{\sim}{T}^{5/3}$. At higher temperatures, there is a broad crossover regime where $\ensuremath{\rho}$ either saturates or $\ensuremath{\rho}\ensuremath{\sim}T$, depending on microscopic details. We discuss these results in the context of recent quantum Monte Carlo simulations of a metallic Ising nematic critical point, and experiments in certain iron-based superconductors.