Topological quantum matter with ultracold gases in optical lattices

Since the discovery of topological insulators, many topological phases have been predicted and realized in a range of different systems, providing both fascinating physics and exciting opportunities for devices. And although new materials are being developed and explored all the time, the prospects for probing exotic topological phases would be greatly enhanced if they could be realized in systems that were easily tuned. The flexibility offered by ultracold atoms could provide such a platform. Here, we review the tools available for creating topological states using ultracold atoms in optical lattices, give an overview of the theoretical and experimental advances and provide an outlook towards realizing strongly correlated topological phases. Using optical lattices to trap ultracold atoms provides a powerful platform for probing topological phases, analogues to those found in condensed matter. But as these systems are highly tunable, they could be used to engineer even more exotic phases. Using optical lattices to trap ultracold atoms provides a powerful platform for probing topological phases, analogues to those found in condensed matter. But as these systems are highly tunable, they could be used to engineer even more exotic phases.