  Localization is one of the most fundamental interference phenomena caused by
randomness, and its universal aspects have been extensively explored from the
perspective of one-parameter scaling mainly in equilibrium states. We
theoretically study dynamics of fermions on disordered one-dimensional
potentials exhibiting localization, and find that surface roughness and
entanglement entropy show dynamical one-parameter-scaling in delocalized
phases. The scaling of the roughness corresponds to the Family-Vicsek scaling
in classical surface growth, and the associated universal scaling exponents
depend on the type of disorder. Particularly, we find that partially localized
states in the delocalized phase of the random-dimer model lead to anomalous
scaling exponents, which are absent in classical systems and clean systems.
Furthermore, we show that the surface roughness is approximately proportional
to the square root of the von Neumann entanglement entropy, and then
demonstrate that even the entanglement entropy obeys the Family-Vicsek-type
scaling. This finding suggests that the surface roughness becomes a reliable
measure for the entanglement dynamics.
