Kagome and honeycomb flat bands in moir\'e graphene

We propose a class of graphene-based moir\'e systems hosting flat bands on kagome and honeycomb moir\'e superlattices. These systems are formed by stacking a graphene layer on a 2D substrate with lattice constant approximately $\sqrt{3}$ times that of graphene. When the moir\'e potentials are induced by a 2D irreducible corepresentation in the substrate, the model shows a rich phase diagram of low energy bands including eigenvalue fragile phases as well as kagome and honeycomb flat bands. Spin-orbit coupling in the substrate can lift symmetry protected degeneracies and create spin Chern bands, and we observe spin Chern numbers up to three. We additionally propose a moir\'e system formed by stacking two graphene-like layers with similar lattice constants and Fermi energies but with Dirac Fermi velocities of opposite sign. This system exhibits multiple kagome and honeycomb flat bands simultaneously. Both models we propose resemble the hypermagic model of [Scheer $\textit{et al.}$, Phys. Rev. B $\textbf{106}$, 115418 (2022)] and may provide ideal platforms for the realization of strongly correlated topological phases.