Symmetry-based classification of exact flat bands in single and bilayer moir\'e systems

We study the influence of spatial symmetries on the appearance and the number of exact flat bands (FBs) in single and bilayer systems with Dirac or quadratic band crossing points, and systematically classify all possible number of exact flat bands in systems with different point group symmetries. We find that a maximum of 6 FBs can be protected by symmetries, and show an example of 6 FBs in a system with QBCP under periodic strain field of $\mathcal{C}_{6v}$ point group symmetry. All known examples of exact FBs in single and bilayer systems fall under this classification, including chiral twisted bilayer graphene, and new examples of exact FBs are found. We show the construction of wavefunctions for the highly degenerate FBs, and prove that any such set of FBs are $\mathds{Z}_2$ nontrivial, where all WFs polarized on one sublattice together have Chern number $C = 1$ and WFs polarized on the other sublattice together have $C = -1$. These bands also satisfy ideal non-Abelian quantum geometry condition. We further show that, just like in TBG, topological heavy fermion description of the FBs with higher degeneracy is possible as long as the Berry curvature distribution is peaked around a point in the BZ.