Perpendicular electric field drives Chern transitions and layer polarization changes in Hofstadter bands

Moir\'e superlattices engineer band properties and enable observation of fractal energy spectra of Hofstadter butterfly. Recently, correlated-electron physics hosted by flat bands in small-angle moir\'e systems has been at the foreground. However, the implications of moir\'e band topology within the single-particle framework are little explored experimentally. An outstanding problem is understanding the effect of band topology on Hofstadter physics, which does not require electron correlations. Our work experimentally studies Chern state switching in the Hofstadter regime using twisted double bilayer graphene (TDBG), which offers electric field tunable topological bands, unlike twisted bilayer graphene. Here we show that the nontrivial topology reflects in the Hofstadter spectra, in particular, by displaying a cascade of Hofstadter gaps that switch their Chern numbers sequentially while varying the perpendicular electric field. Our experiments together with theoretical calculations suggest a crucial role of charge polarization changing concomitantly with topological transitions in this system. Layer polarization is likely to play an important role in the topological states in few-layer twisted systems. Moreover, our work establishes TDBG as a novel Hofstadter platform with nontrivial magnetoelectric coupling.