The importance of the quantum metric in flat-band systems has been noticed
recently in many contexts such as the superfluid stiffness, the dc electrical
conductivity, and ideal Chern insulators. Both the quantum metric of degenerate
and nondegenerate bands can be naturally described via the geometry of
different Grassmannian manifolds, specific to the band degeneracies. Contrary
to the (Abelian) Berry curvature, the quantum metric of a degenerate band
resulting from the collapse of a collection of bands is not simply the sum of
the individual quantum metrics. We provide a physical interpretation of this
phenomenon in terms of transition dipole matrix elements between two bands. By
considering a toy model, we show that the quantum metric gets enhanced,
reduced, or remains unaffected depending on which bands collapse. The dc
longitudinal conductivity and the superfluid stiffness are known to be
proportional to the quantum metric for flat-band systems, which makes them
suitable candidates for the observation of this phenomenon.
