Multi-scale lattice relaxation in chiral twisted trilayer graphenes

We theoretically investigate the lattice relaxation and the electronic property in non-symemtric chiral TTGs by using an effective continuum model. The relaxed lattice structure forms a patchwork of moir\'e-of-moir\'e domains, where the moir\'e patterns given by layer 1 and 2, and layer 2 and 3 become locally commensurate with a specific relative alignment. The band calculation reveals a wide energy window (> 50 meV) with low density of states, featuring sparsely distributed highly one-dimensional electron bands. The wave function of these one-dimensional bands exhibits sharp localization at the boundaries between super-moir\'e domains. By calculating the Chern number of the local band structure within commensurate domains, the one-dimensional state is identified as a topological boundary state between distinct Chern insulators.