Replica Higher-Order Topology of Hofstadter Butterflies in Twisted Bilayer Graphene

TL;DR

This paper reveals that twisted bilayer graphene exhibits recursive higher-order topological insulator phases within its Hofstadter spectrum, demonstrating self-similarity and symmetry-protected topological features in quantum fractals.

Contribution

It uncovers the existence of replica HOTI phases in Hofstadter butterflies of twisted bilayer graphene, extending the understanding of topological properties in fractal energy spectra.

Findings

Replica HOTIs occur at various fluxes without protecting symmetries.

Original HOTI phase re-appears at half-flux periodicity.

Corner states and topological markers are present in replicas.

Abstract

The Hofstadter energy spectrum of twisted bilayer graphene 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 the exact flux translational symmetry of twisted bilayer graphene at all commensurate angles. Based on this result, we carefully 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. Similar to 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 twisted bilayer graphene. The topological aspect of Hofstadter butterflies revealed in our results highlights symmetry-protected topology in quantum fractals.