From Charge Fluctuations to Geometry and Topology of Quantum Matter

Charge fluctuations provide a powerful and experimentally accessible window into correlations and entanglement in quantum matter. In this talk, I will show how carefully chosen charge-fluctuation measurements reveal both the quantum geometric and topological structure of insulators and metals.

In the first part, I will explain how the corner contribution to bipartite charge fluctuations encodes the integrated quantum metric in generic lattice insulators [1]. The quantum metric is a central object in quantum geometry, characterizing wavefunction localization and obeying a topological bound. This connection further motivates a unified perspective linking quantum information and quantum geometry via the lens of corner entanglement entropy.

In the second part, I will show how multipartite charge fluctuations—equivalently, higher-order density correlations—capture the topology of metals [2,3]. In particular, the topology of a metallic state can be classified by the Euler characteristic of its Fermi sea, and in D spatial dimensions this invariant is encoded in the (D+1)-point density correlation or (D+1)-partite charge fluctuation. For Fermi gases, I will present a recent quantum-gas-microscopy experiment that directly measured the predicted topological density correlations in D=2, providing the first experimental verification of this framework. If time permits, I will also discuss how Fermi-surface geometry can be probed using charge fluctuations.

Overall, these results demonstrate that charge fluctuations form a versatile and experimentally practical probe that unifies quantum geometry, topology, and entanglement across a wide range of quantum materials.

[1] Phys. Rev. Lett. 133, 246603 (2024); [2] Phys. Rev. X 12, 031022 (2022); [3] Phys. Rev. B 109, 035413 (2024)