  Quench dynamics in a two-dimensional system of interacting fermions is
analyzed within the semiclassical truncated Wigner approximation (TWA). The
models with short-range and long-range interactions are considered. We show
that in the latter case, the TWA is very accurate, becoming asymptotically
exact in the infinite-range limit, provided that the semiclassical Hamiltonian
is correctly identified. Within the TWA, different dynamical timescales of
charges and spins can be clearly distinguished. Interestingly, for a weak and
moderate disorder strength, we observe subdiffusive behavior of charges, while
spins exhibit diffusive dynamics. At strong disorder, the quantum Fisher
information shows logarithmic growth in time with a slower increase for charges
than for spins. It is shown that in contrast to the short-range model, strong
inhomogeneities such as domain walls in the initial state can significantly
slow down thermalization dynamics, especially at weak disorder. This behavior
can put additional challenges in designing cold-atom experimental protocols
aimed to analyze possible many-body localization in such systems. While within
this approach we cannot make any definite statements about the existence of a
many-body localized phase, we see a very fast crossover as a function of
disorder strength from rapidly thermalizing to a slow glassy like regime both
for the short-range and long-range models.
