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Localization is one of the most fundamental interference phenomena caused byrandomness, and its universal aspects have been extensively explored from theperspective of one-parameter scaling mainly for static properties. Wenumerically study dynamics of fermions on disordered onedimensional potentialsexhibiting localization and find dynamical one-parameter scaling for surfaceroughness, which represents particle-number fluctuations at a givenlengthscale, and for entanglement entropy when the system is in delocalizedphases. This dynamical scaling corresponds to the Family-Vicsek scalingoriginally developed in classical surface growth, and the associated scalingexponents depend on the type of disorder. Notably, we find that partiallylocalized states in the delocalized phase of the random-dimer model lead toanomalous scaling, where destructive interference unique to quantum systemsleads to exponents unknown for classical systems and clean systems.

Dynamical Scaling of Surface Roughness and Entanglement Entropy in Disordered Fermion Models