  We propose and analyze a scheme to observe topological phenomena with ions in
microtraps. We consider a set of trapped ions forming a regular structure in
two spatial dimensions and interacting with lasers. We find phonon bands with
non-trivial topological properties, which are caused by the breaking of time
reversal symmetry induced by the lasers. We investigate the appearance of edge
modes, as well as their robustness against perturbations. Long-range hopping of
phonons caused by the Coulomb interaction gives rise to flat bands which,
together with induced phonon-phonon interactions, can be used to produce and
explore strongly correlated states. Furthermore, some of these ideas can also
be implemented with cold atoms in optical lattices.
