  The quantum metric is a central quantity of band theory but has so far not
been related to many response coefficients due to its nonclassical origin.
However, within a newly developed Kubo formalism for fast relaxation, the
decomposition of the dc electrical conductivity into both classical (intraband)
and quantum (interband) contributions recently revealed that the interband part
is proportional to the quantum metric. Here, we show that interband effects due
to the quantum metric can be significantly enhanced and even dominate the
conductivity for semimetals at charge neutrality and for systems with highly
quenched bandwidth. This is true in particular for topological flat-band
materials of nonzero Chern number, where for intermediate relaxation rates an
upper bound exists for the resistivity due to the common geometrical origin of
quantum metric and Berry curvature. We suggest to search for these effects in
highly tunable rhombohedral trilayer graphene flakes.
