Abstract 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.
Twisted bilayer graphene: Replica higher-order topology of Hofstadter butterflies
Magnetic translational symmetry of crystals in the presence of an external magnetic field manifests as a fractal form of the energy spectrum that resembles recurring replicas of butterflies, known as Hofstadter butterflies. The magnetic field required to produce replicas of the Landau levels could be significantly reduced by the large-scale synthesis of a van der Waals superlattice with a macroscopic unit cell. For this crucial development, the Hofstadter butterflies have been experimentally realized in a graphene superlattice, magic-angle twisted bilayer graphene (TBG), and twisted double-bilayer graphene. Notably, the link with the magnetic translational symmetry and symmetry-protected topological phases of matter has been revealed recently. In a general lattice model with multiple sites per unit cell, the Hofstadter energy spectrum becomes approximately replicative under the addition of the flux periodicity, which constitutes the additional flux translational symmetry via the unitary transformation of the Hamiltonian. Remarkably, the effective time-reversal symmetry is restored at a half-flux periodicity, allowing for the existence of diverse topological states of matter protected by symmetries.In this work, Sun-Woo Kim et al. from the Department of Physics, KAIST, Korea,demonstrated 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. They showed the existence of exact flux translational symmetry in TBG at all commensurate angles. Based on this result, they identified 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 were 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 these results highlights symmetry-protected topology in quantum fractals. This result can pave the way for studying replica topology under magnetic field in generic moiré multilayer and moiré quasiperodic systems that host multiple interaction scales.
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