Nontrivial quantum geometry of degenerate flat bands

TL;DR

This paper explores the complex quantum geometry of degenerate flat bands, revealing how the quantum metric behaves differently from nondegenerate bands and impacts physical properties like conductivity and superfluid stiffness.

Contribution

It provides a new physical interpretation of the quantum metric in degenerate bands and demonstrates how band collapse affects this metric through a toy model.

Findings

Quantum metric of degenerate bands differs from the sum of individual metrics.

Band collapse can enhance, reduce, or leave the quantum metric unchanged.

Physical properties like conductivity and superfluid stiffness are linked to the quantum metric.

Abstract

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.