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The importance of the quantum metric in flat-band systems has been noticedrecently in many contexts such as the superfluid stiffness, the dc electricalconductivity, and ideal Chern insulators. Both the quantum metric of degenerateand nondegenerate bands can be naturally described via the geometry ofdifferent Grassmannian manifolds, specific to the band degeneracies. Contraryto the (Abelian) Berry curvature, the quantum metric of a degenerate bandresulting from the collapse of a collection of bands is not simply the sum ofthe individual quantum metrics. We provide a physical interpretation of thisphenomenon in terms of transition dipole matrix elements between two bands. Byconsidering a toy model, we show that the quantum metric gets enhanced,reduced, or remains unaffected depending on which bands collapse. The dclongitudinal conductivity and the superfluid stiffness are known to beproportional to the quantum metric for flat-band systems, which makes themsuitable candidates for the observation of this phenomenon.

Nontrivial quantum geometry of degenerate flat bands