Nontrivial quantum geometry of nondegenerate and degenerate bands and its implications for flat-band systems

The importance of quantum geometry and, in particular, of the quantum metric has been identified throughout various fields in the recent years. Especially flat-band systems are promising candidates for new relevant quantum metric effects, since well-known contributions to observables that depend on the slope or curvature of the dispersion vanish. For instance, the superfluid stiffness and the dc electrical conductivity yield dominant contributions proportional to the integrated quantum metric for flat bands. In this talk, I will address the role of band degeneracy for these quantum metric effects. I will discuss how the quantum metric of both degenerate and nondegenerate bands can be naturally described via the geometry of different Grassmannian manifolds, specific to the band degeneracies. An interesting property is the non-additivity of the quantum metric under the collapse of a collection of bands. In contrast, the Berry curvature is additive. We will discuss this effect for a non-trivial three-band model, where the quantum metric gets enhanced, reduced, or remains unaffected depending on which bands collapse. I will use the developed framework for dc electrical conductivity [1,2] to discuss differences and similarities when bands are nondegenerate or degenerate, which can be directly applied and extended to other observables.

[1] Mitscherling and Holder, PRB 105, 085154 (2022)

[2] Mera and Mitscherling, PRB 106, 165133 (2022)