Periodic Table of Flat Bands and the Origin of Band Flatness

Flat bands provide a natural platform for emergent electronic states beyond Landau paradigm. However, it is still considered a great "luck" to find a new flat band. Sometimes we can do it through careful engineering (fine-tuning") of material properties, sometimes we can do it by twisting multilayer heterostructures, sometimes we involve magnetic fields to produce flat Landau levels. Is there something similar between these different cases of flat bands? It appears that in each of these cases we can point out on the common origin of the perfectly flat bands, which is the (self)-trapping in the real space. To find the essence of the phenomena, we investigate only the perfectly flat bands, which surprisingly form a "periodic table" depending on their topological and tight binding properties. The claim is that all the realistic flat band are derivatives of (7+1) classes of perfectly flat bands on the lattice, including radically different examples such as atomic insulator, Landau levels, and perfectly flat bands in twisted bilayer graphene. Through using the quantum geometric tensor we justify the remarkable connection between band flatness obstruction and the (higher) Chern numbers, an argument which is independent from the lattice symmetries.