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

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 $\mathrm{\mid }C\mathrm{\mid }=2$ 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 $\mathit{\text{bosonic}}$, characterized by electronic charge $Q=2e$, that is $\mathit{\text{even}}$ 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 $\pm \frac{2e^2}{h},\pm \frac{4e^2}{h},\mathrm{.}\mathrm{.}\mathrm{.}$ for TBG with skyrmion order.