Replica higher-order topology of Hofstadter butterflies in twisted bilayer graphene

The Hofstadter energy spectrum of twisted bilayer graphene (TBG) is found to have recursive higher-order topological properties. We demonstrate that higher-order topological insulator (HOTI) phases, characterized by localized corner states, occur as replicas of the original HOTIs to fulfill the self-similarity of the Hofstadter spectrum. We show the existence of exact flux translational symmetry in TBG at all commensurate angles. Based on this result, we identify that the original HOTI phase at zero flux is re-entrant at a half-flux periodicity, where the effective twofold rotation is preserved. In addition, numerous replicas of the original HOTIs are found for fluxes without protecting symmetries. Like the original HOTIs, replica HOTIs feature both localized corner states and edge-localized real-space topological markers. The replica HOTIs originate from the different interaction scales, namely, intralayer and interlayer couplings, in TBG. The topological aspect of Hofstadter butterflies revealed in our results highlights symmetry-protected topology in quantum fractals.