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

We calculate the electronic structure in quasiperiodic double moiré systems of graphene sandwiched by hexagonal boron nitride (hBN) and identify the topological integers of energy gaps. We find that the electronic spectrum contains a number of minigaps, and they exhibit a recursive fractal structure similar to the Hofstadter butterfly when plotted against the twist angle. Each of the energy gaps can be characterized by a set of six integers, which are associated with an area in momentum space. The corresponding area is geometrically interpreted as a quasi-Brillouin zone, which is a polygon enclosed by multiple Bragg planes of the composite periods and can be uniquely specified by the plain wave projection in the weak-potential limit. These sets of integers can be expressed by the second Chern numbers by using a formal mapping to 4D quantum Hall effect.