Flat bands and long range Coulomb interactions: conducting or insulating?

Dispersionless (flat) electronic bands are investigated regarding their conductance properties. Due to "caging" of carriers these bands are usually insulating at partial filling, at least on the non-interacting level. Considering the specific example of a $\mathcal{T}_3$--lattice we study long-range Coulomb interactions. A non-trivial dependence of the conductivity on flat band filling is obtained, exhibiting an infinite number of zeros. Near these zeros, the conductivity rises linearly with carrier density. At densities half way in between adjacent conductivity-zeros, strongly enhanced conductivity is predicted, accompanying a solid-solid phase transition.