Realizing and detecting topological phases in momentum-space lattices

We propose feasible schemes to realize topological phases with cold atoms in momentum-space lattices through carefully engineering artificial gauge potentials for neutral atoms. We show a Floquet approach to design momentum-space lattices with long-range hoppings, which can be applicable to realize the momentum-space Haldane model. We further demonstrate the realization of the two-dimensioanl spin-orbit coupling for a spinful momentum-space lattices, arising from a general formulation of non-Abelian gauge potential. Then we give a minimal realization describing quantum anomalous Hall effect in a spin-orbit-coupling induced Chern insulator. We also provide three different dynamical approaches to extract topological properties of the proposed models. The numerical results show that our approaches are still valid even for a lattice with small size and in a short evolution time, thus making it feasible for realistic experiments.