Electric field drives Chern transition in Hofstadter bands of twisted double bilayer graphene*

Moiré superlattices engineer band properties and enable observation of fractal energy spectra of Hofstadter butterfly. Recently, a major focus has been to gain new insights into correlated-electron physics hosted by the flat bands in small-angle moiré systems. However, the topological properties of these moiré bands such as Chern numbers are little explored. We study the effect of band topology on Hofstadter physics using twisted double bilayer graphene (TDBG), where the perpendicular electric field tunes the band structure and its topological properties. In addition to the signatures of nontrivial topology in the Hofstadter spectra, we observe a cascade of Chern gaps that switch their Chern numbers one-at-a-time as we tune the electric field. Our experiments together with theoretical calculations suggest a crucial role of charge polarization in TDBG that changes concomitantly with topological transitions. The layer polarization is likely to play an important role in the topological states in few-layer twisted systems. Overall, our work establishes TDBG as a novel Hofstadter platform to switch magnetic moment tunable with an external knob of the electric field.