Corner Charge Fluctuation as an Observable for Quantum Geometry and Entanglement in Two-dimensional Insulators

TL;DR

This paper demonstrates that corner charge fluctuations in 2D insulators reveal quantum geometry and entanglement properties, providing a new experimental probe for understanding topological and quantum information aspects.

Contribution

It establishes a direct relation between corner charge fluctuations and quantum geometry, offering a practical scheme and analytical proof for their angle dependence in lattice systems.

Findings

Corner charge fluctuation's angle dependence measures the quantum metric.

Numerical verification confirms the relevance for Chern insulators.

Connection between quantum geometry and entanglement entropy for free fermions.

Abstract

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. Here we establish that, for generic lattice systems of interacting particles, the corner charge fluctuation is directly related to quantum geometry. 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. A model of a compact obstructed atomic insulator is introduced to illustrate this effect analytically, while numerical verification for various Chern insulator models further demonstrate the experimental relevance of the corner charge fluctuation in a finite-size quantum simulator as a probe of quantum geometry. Last but not least, for free fermions, we unveil an intimate connection between quantum geometry and quantum information through the lens of corner entanglement entropies.