  Flat bands provide a natural platform for emergent electronic states beyond
Landau paradigm. Among those of particular importance are flat Chern bands,
including bands of higher Chern numbers ($C$$>$$1$). We introduce a new
framework for band flatness through wave functions, and classify the existing
isolated flat bands in a "periodic table" according to tight binding features
and wave function properties. Our flat band categorization encompasses
seemingly different classes of flat bands ranging from atomic insulators to
perfectly flat Chern bands and Landau Levels. The perfectly flat Chern bands
satisfy Berry curvature condition $F_{xy} = \text{Tr} \, \mathcal G_{ij}$ which
on the tight-binding level is fulfilled only for infinite-range models. Most of
the natural Chern bands fall into category of $C=1$; the complexity of creating
higher-$C$ flat bands is beyond the current technology. This is due to the
breakdown of the microscopic stability for higher-$C$ flatness, seen
atomistically e.g. in the increase of the hopping range bound as
$\propto$$\sqrt{C} a$. Within our new formalism, we indicate strategies for
bypassing higher-$C$ constraints and thus dramatically decreasing the
implementation complexity.
