Quantum geometry and quantum entanglement: a view from the corner

Measuring bipartite fluctuations of a conserved charge, such as the particle number, is a powerful approach to understanding quantum systems. When the measured region has sharp corners, the bipartite fluctuation receives an additional contribution known to exhibit a universal angle-dependence in 2D isotropic and uniform systems. In this talk we establish that, for generic lattice systems of interacting particles, the corner charge fluctuation directly probes the quantum geometry characterizing the many-body insulating ground state. We first provide a practical scheme to isolate the corner contribution on lattices, and analytically prove that its angle-dependence in the small-angle limit measures exclusively the integrated quantum metric. We further present numerical verification for various Chern insulator models to demonstrate the experimental relevance of the corner charge fluctuation in a finite-size quantum simulator as a practical probe of quantum geometry. Last but not least, for free fermions, we unveil an intimate quantitative connection between quantum geometry and quantum information through the lens of corner entanglement entropies.