Bound on resistivity in flatband materials due to the quantum metric

Recently, the quantum metric attracted a lot of attention by providing a lower bound to the superfluid stiffness of the superconducting state in flat-band systems. However until now, this central quantity of band theory could not be related to many response coefficients due to its non-classical origin. In this talk, I show that the linear electrical conductivity involves a new interband contribution based on the quantum metric. In systems with highly quenched bandwidth this quantum metric contribution can even dominate the longitudinal conductivity. Furthermore, for topological flat bands with nonzero Chern number the quantum metric contribution is bounded from below, i.e. I propose a new geometric bound on the longitudinal resistivity. These effects might be observable in highly tunable rhombohedral trilayer graphene flakes.