Timescale for macroscopic equilibration in isolated quantum systems: a rigorous derivation for free fermions

For a class of translation-invariant free-fermion systems (including those with uniform nearest neighbor hopping) on a $d$-dimensional $L \times \cdots \times L$ hypercubic lattice, we prove that, starting from an arbitrary pure initial state, the system equilibrates with respect to the coarse-grained density within a timescale of order $L$. This scaling is optimal, since there exist initial states whose equilibration requires time of order $L$. Our result establishes $O(L)$ as the equilibration timescale, as is expected in normal macroscopic systems with a conserved quantity, such as total number of particles.