Semiclassical bounds on dynamics of two-dimensional interacting disordered fermions

Using the truncated Wigner approximation (TWA) we study quench dynamics of two-dimensional lattice systems consisting of interacting spinless fermions with potential disorder. First, we demonstrate that the semiclassical dynamics generally relaxes faster than the full quantum dynamics. We obtain this result by comparing the semiclassical dynamics with exact diagonalization and Lanczos propagation of one-dimensional chains. Next, exploiting the TWA capabilities of simulating large lattices, we investigate how the relaxation rates depend on the dimensionality of the studied system. We show that strongly disordered one-dimensional and two-dimensional systems exhibit a transient, logarithmic-in-time relaxation, which was recently established for one-dimensional chains. Such relaxation corresponds to the infamous $1/f$-noise at strong disorder.