  Flatband systems form a new class of materials with seemingly contradictory
properties that challenge our understanding of their transport phenomenology.
On the one hand, the large density of states at the chemical potential suggests
a metallic behavior. On the other hand, all states have a vanishing velocity
that essentially prevents transport. Furthermore, electronic correlation
effects are intrinsically strong as the interaction energy scale naturally
exceeds the narrow or vanishing bandwidth. In this work, we determine the
conductivity of flatband systems by investigating their Drude weight, thereby
avoiding issues associated with a particular choice of relaxation broadening.
Our study is a controlled analytic treatment performed at temperatures above
the bandwidth and the interaction strength and below the bandgap, a new
distinctive transport regime for flat bands. We find that the Drude weight has
geometric contributions that are unique to a perfectly flat band and are beyond
the commonly studied quantum metric and Berry curvature. We expect that these
novel contributions are essential in determining flatband response functions
and shed new light on the stability of flatband systems.
