  Charge transport in disordered two-dimensional (2D) systems showcases a
myriad of unique phenomenologies that highlight different aspects of the
underlying quantum dynamics. Electrons in such systems undergo a crossover from
ballistic propagation to Anderson localization, contingent on the system's
effective coherence length. Between the extended and localized phases lies a
diffusive crossover in which the charge conductivity is properly defined. The
numerical observation of this regime has remained elusive because it requires
fully coherent transport to be simulated in systems whose dimensions are
sufficiently large to meaningfully split the mean-free path and localization
length scales. To address this challenge, we employed a novel linear scaling
time-resolved approach that enabled us to derive the dc-transport
characteristics and observe the three expected 2D transport regimes -
ballistic, diffusive, and localized.
