Helical trilayer graphene in magnetic field: Chern mosaic and higher Chern number ideal flat bands

Helical trilayer graphene (hTG) exhibits a supermoir\'e pattern with large domains centered around stacking points ABA and BAB, where two well-separated low-energy bands appear with different total Chern numbers at each valley, forming a Chern mosaic pattern. In the chiral limit, the low-energy bands become exactly flat at zero energy for magic-angle twists. Here we investigate these zero-energy flat bands and their topological properties in the presence of a perpendicular magnetic field. We show that hTG retains the precise flatness of the zero-energy bands, even at finite magnetic fields. We find topological phase transitions at fields corresponding to unit and half magnetic flux leading to an emergence of higher Chern number flat bands. Consequently the Chern mosaic gets modified for finite magnetic fields. We further find the analytical forms of zero-energy wave functions and identify a set of hidden wave functions, which gives crucial insights into both the topological transitions and enhancement of Chern numbers across them. We also find topological transitions away from the chiral limit with finite corrugations and at different magic angles.