Unconventional Flat Chern Bands and 2$e$ Charges in Skyrmionic Moir\'e Superlattices

Abstract

The interplay of topological characteristics in real space and reciprocal space can lead to the emergence of unconventional topological phases. In this Letter, we implement a novel mechanism for generating higher-Chern flat bands on the basis of twisted bilayer graphene (TBG) coupled to topological magnetic structures in the form of the skyrmion lattice. In particular, we discover a scenario for generating $|C|=2$ dispersionless electronic bands when the skyrmion periodicity and the moir\'e periodicity are matched. Following the Wilczek argument, the statistics of the charge-carrying excitations in this case is \textit{bosonic}, characterized by electronic charge $Q =2e$, that is \textit{even} in units of electron charge $e$. The required skyrmion coupling strength triggering the topological phase transition is realistic, with its threshold estimated as low as 4~meV. The Hofstadter butterfly spectrum of this phase is different resulting in an unexpected quantum Hall conductance sequence $\pm \frac{2 e^2}{h}, \ \pm \frac{4 e^2}{h}, ...$ for TBG with skyrmion order.