  We propose a realistic scheme to construct anomalous Floquet Chern
topological insulators using spin-1/2 particles carrying out a discrete-time
quantum walk in a two-dimensional lattice. By Floquet engineering the
quantum-walk protocol, an Aharonov-Bohm geometric phase is imprinted onto
closed-loop paths in the lattice, thus realizing an abelian gauge field---the
analog of a magnetic flux threading a two-dimensional electron gas. We show
that in the strong field regime, when the flux per plaquette is a sizable
fraction of the flux quantum, magnetic quantum walks give rise to nearly flat
energy bands featuring nonvanishing Chern numbers. Furthermore, we find that
because of the nonperturbative nature of the periodic driving, a second
topological number---the so-called RLBL invariant---is necessary to fully
characterize the anomalous Floquet topological phases of magnetic quantum walks
and to compute the number of topologically protected edge modes expected at the
boundaries between different phases. In the second part of this article, we
discuss an implementation of this scheme using neutral atoms in two-dimensional
spin-dependent optical lattices, which enables the generation of arbitrary
magnetic-field landscapes, including those with sharp boundaries. The robust
atom transport, which is observed along boundaries separating regions of
different field strength, reveals the topological character of the Floquet
Chern bands.
