Quantum-geometric bounds on electronic properties

Quantum-geometric quantities satisfy rigorous bound relations among themselves, and with other electronic properties such as the energy gap and the electric susceptibility. Some of those relations can be linked, via sum rules, to the optical absorption spectrum. In this talk I will discuss bound relations involving the Berry curvature, quantum metric, orbital moment, and inverse effective mass, providing illustrative applications to model systems and real materials. The impact of long-ranged van der Waals forces on the many-electron localization (ground-state quantum metric) will be illustrated via analytically solvable harmonic oscillator models. The manner in which certain inequalities approach saturation in a model Chern insulator with tunable flat bands will be analyzed in terms of the optical absorption spectrum. Finally, the metric-curvature inequality will be analyzed for an ab-inito-based model of gapped graphene.