The dynamics of a one-dimensional two-component Fermi gas in the presence of a quasi-periodic optical lattice (OL) is investigated by means of a Density Functional Theory approach. Inspired by the protocol implemented in recent cold-atom experiments, designed to identify the many-body localization transition, we analyze the relaxation of an initially prepared imbalance between the occupation number of odd and of even sites. For quasi-disorder strength beyond the Anderson localization transition, the imbalance survives for long times, indicating the inability of the system to reach local equilibrium. The late time value diminishes for increasing interaction strength. Close to the critical quasi-disorder strength corresponding to the noninteracting (Anderson) transition, the interacting system displays an extremely slow relaxation dynamics, consistent with sub-diffusive behavior. The amplitude of the imbalance fluctuations around its running average is found to decrease with time, and such damping is more effective with increasing interaction strengths. While our study addresses the setup with two equally intense OLs, very similar effects due to interactions have been observed also in recent cold-atom experiments performed in the tight-binding regime, i.e. where one of the two OLs is very deep and the other is much weaker.
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