The Drude weight of a flatband metal

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.