Topological exact flat bands in moiré materials with quasiperiodic potentials

Electronic states in quasicrystalline systems offer rich and complex phenomena. Since they occur rarely in nature, one way to engineer quasiperiodicity is by tuning moiré potential in 2D materials. Motivated by recent experimental progress in tuning moiré quasiperiodic systems, we study occurence of exact flat bands in systems with quadratic band crossing points at chiral limit under moiré potential containing multiple harmonics as an approximation to quasiperiodicity. We show the existence of critical surfaces, where exact flat bands occur, in the space of amplitudes of different harmonics. We find that the co-dimensions of these surfaces depend on the point group symmetry of the moiré potential. We further investigate how a critical point for one harmonic splits into several critical surfaces under perturbative addition of other harmonics, and give exact counting rules and corresponding flat band degeneracy for the critical surfaces that the critical point splits into. We comment on the topology and quantum geometry of the flat bands.