  Motivated by recent advances in the realization of complex two-dimensional
optical lattices, we investigate theoretically the quantum transport of
ultracold fermions in an optical kagome lattice. In particular, we focus on its
extensively degenerate localized states (flat band). By loading fermions in a
partial region of the lattice and depleting the mobile atoms at the far
boundary of the initially unoccupied region, we find a dynamically generated
flat-band insulator, which is also a population-inverted state. We further show
that inclusion of weak repulsion leads to a dynamical stripe phase for
two-component fermions in a similar setup. Finally, by preparing a topological
insulating state in a partially occupied kagome lattice, we find that the
topological chiral current decays but exhibits an interesting oscillating
dynamics during the nonequilibrium transport. Given the broad variety of
lattice geometries supporting localized or topological states, our work
suggests new possibilities to use geometrical effects and their dynamics in
atomtronic devices.
