Fractal energy gaps and topological invariants in hBN/graphene/hBN double moiré systems

We calculate the electronic structure in quasiperiodic double-moir\'e systems of graphene sandwiched by hexagonal boron nitride, and identify topological invariants energy gaps. find that spectrum contains a number minigaps, they exhibit recursive fractal similar to Hofstadter butterfly when plotted against twist angle. Each gaps can be characterized set integers, which are associated with an area momentum space. The corresponding is geometrically interpreted as quasi Brillouin zone, polygon enclosed multiple Bragg planes composite periods uniquely specified plain wave projection weak potential limit.