Interaction Corrections to the Quantum Metric

The quantum metric — the real part of the quantum-geometric tensor — quantifies the amplitude distance between nearby quantum states and complements the Berry curvature’s measure of phase winding. Long treated as secondary, the quantum metric has started gaining prominence due to its recently discovered role in transport, optical responses and superconductivity. While most applications to date use single-band (or band-projected) metrics, interacting systems call for a many-body quantum metric. Recent work shows that the many-body quantum metric is directly encoded in the long-wavelength static structure factor through the “quantum weight”, thus opening a concrete route to measure or compute the many-body metric. This connection elevates density fluctuations to a primary probe of quantum geometry in correlated phases.  Motivated by this program, we compute the density–density correlation function χ(q, iΩ_n) for the spinless Haldane model with density–density interactions using a Matsubara-frequency, diagrammatic approach. From χ we extract the static structure factor S(q) and its small-q expansion.