  We study the electrical transport of a two-dimensional non-Fermi liquid with
disorder, and we determine the first quantum correction to the semiclassical dc
conductivity due to quantum interference. We consider a system with $N$ flavors
of fermions coupled to SU($N$) critical matrix bosons. Motivated by the SYK
model, we employ the bilocal field formalism and derive a set of
finite-temperature saddle-point equations governing the fermionic and bosonic
self-energies in the large-$N$ limit. Interestingly, disorder smearing induces
a marginal Fermi liquid (MFL) self-energy for the fermions. We next consider
fluctuations around the saddle points and derive a MFL-Finkel'stein nonlinear
sigma model. We find that the Altshuler-Aronov quantum conductance correction
gives linear-$T$ resistivity that can dominate over the Drude result at low
temperature. The strong temperature dependence of the quantum correction arises
due to rapid relaxation of the mediating quantum-critical bosons. We verify
that our calculations explicitly satisfy the Ward identity at the semiclassical
and quantum levels. Our results establish that quantum interference persists in
two-particle hydrodynamic modes, even when quasiparticles are subject to strong
(Planckian) dissipation.
