Kagome flat bands, fragile topology, quantum spin Hall effect, and more in graphene moire materials

We present a moire continuum model for graphene modulated by moire interference with another two-dimensional material. Without considering spin-orbit coupling, the model exhibits a rich phase diagram including both fragile topological bands and bands with trivial symmetry indicators. Many elementary and composite band representations of the space group P6mm appear as the symmetry indicators for connected bands near charge neutrality. Examples include one-orbital kagome lattice bands and two-orbital honeycomb lattice bands, both of which include flat bands. If one includes spin-orbit coupling, quantum spin Hall bands emerge with spin Chern numbers up to 3. Experimental realizations of this model may lead to a deeper understanding of correlations in flat bands and topological phases of matter.