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.

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