Nontrivial quantum geometry of degenerate flat bands

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.

Figure 1

The longitudinal conductivity ${\sigma }^{xx}$ of a flat band is proportional to the corresponding integrated quantum metric ${\overline{g}}_f^{xx}$ (dashed lines). When the bands become degenerate, we find a pronounced drop for $\mu ={\varepsilon }_0$ (a) and peak for $\mu ={\varepsilon }_{+}$ (b). In the inset, we show the energy levels of the three bands.

Figure 2

The longitudinal conductivity ${\sigma }^{xx}$ as a function of a phenomenological band broadening $\mathrm{\Gamma }$ for different relative position of the three bands. We understand the value of the observed plateaus (dashed lines) by the quantum metric of the involved one, two, or three bands (inset). The crossovers are given by the band gaps (vertical lines).