Topological and stacked flat bands in bilayer graphene with a superlattice potential

We show that bilayer graphene in the presence of a 2D superlattice potential provides a highly tunable setup that can realize a variety of flat band phenomena. We focus on two regimes: (i) topological flat bands with non-zero Chern numbers, C, including bands with higher Chern numbers |C| > 1; and (ii) an unprecedented phase consisting of a stack of nearly flat bands with C = 0. For realistic values of the potential and superlattice periodicity, this stack can span nearly 100 meV, encompassing nearly all of the low-energy spectrum. Our results provide a realistic guide for future experiments to realize a new platform for flat band phenomena.