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The dynamics of a one-dimensional two-component Fermi gas in the presence ofa quasi-periodic optical lattice (OL) is investigated by means of a DensityFunctional Theory approach. Inspired by the protocol implemented in recentcold-atom experiments, designed to identify the many-body localizationtransition, we analyze the relaxation of an initially prepared imbalancebetween the occupation number of odd and of even sites. For quasi-disorderstrength beyond the Anderson localization transition, the imbalance survivesfor long times, indicating the inability of the system to reach localequilibrium. The late time value diminishes for increasing interactionstrength. Close to the critical quasi-disorder strength corresponding to thenoninteracting (Anderson) transition, the interacting system displays anextremely slow relaxation dynamics, consistent with sub-diffusive behavior. Theamplitude of the imbalance fluctuations around its running average is found todecrease with time, and such damping is more effective with increasinginteraction strengths. While our study addresses the setup with two equallyintense OLs, very similar effects due to interactions have been observed alsoin 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.

Out-of-equilibrium dynamics of repulsive Fermi gases in quasi-periodic potentials: a Density Functional Theory study