Fundamental Tests of Quantum Geometric Bounds in Ionic and Covalent Insulators using Inelastic X-Ray Scattering

TL;DR

This study demonstrates that inelastic x-ray scattering can directly quantify quantum geometry in insulators by extracting the quantum Fisher information and the longitudinal Bures metric from density fluctuations. By analyzing diamond and LiF, the authors relate a dimensionless quantum weight $aK(\mathbf{q})$ to bonding character, showing covalent diamond exhibits greater electronic delocalization and entanglement than ionic LiF. They derive wavevector-dependent bounds on $K(\mathbf{q})$ using the $f$-sum rule and Kramers–Kronig relations, with bounds expressed in terms of the gap $E_g$, static dielectric function $\epsilon(\mathbf{q})$, and plasma frequency $\omega_p$. The results connect quantum information to chemical bonding and validate IXS as a powerful tool to experimentally access quantum geometry in solids, including potential extensions to metallic systems and entanglement-based constraints. The work also links the quantum weight to entanglement through tight-binding analyses, highlighting correlation effects as a source of enhanced information density in covalent networks.

Abstract

Quantum geometry underlies many fundamental properties of materials, but it has remained largely inaccessible to direct experiment. Here we demonstrate that inelastic x-ray scattering (IXS) provides a direct, quantitative probe of quantum geometry and quantum information in solids. Studying two prototype insulators, covalently bonded diamond and ionically bonded LiF, we measure the density response and experimentally determine the quantum Fisher information, the associated Bures metric, and the electron localization length. These measurements enable a quantitative comparison of quantum geometry for two distinct bonding environments. We find that the dimensionless quantum weight, $aK(q)$, which quantifies the longitudinal localization of quantum information, is constrained by fundamental electrostatic bounds in both materials. Crucially, the quantum weight of diamond exceeds that of LiF, indicating that covalent bonds exhibit a higher degree of delocalization and higher density of quantum information than the ionic bonds. Our results establish a direct experimental relationship between quantum information, electron localization, and chemical bonding, and identify IXS as a powerful tool for measuring quantum geometry in materials.

Figures (7)

• Figure 1: IXS spectra of diamond (top) and LiF (bottom) for various momentum transfer. The quasi-elastic peak, corresponding to data below $3 \ eV$, have been set to zero. The vertical lines correspond to the $q \rightarrow 0$ value of the band gap as reported in onishiQuantumWeightFundamental2025.
• Figure 2: Value of the $f$-sum rule integral [Eq. \ref{['eq: f-sum rule']}] for the scaled IXS data for (a) diamond and (b) LiF, plotted against the momentum transfer. The dashed line represents the right hand side of Eq. \ref{['eq: f-sum rule']} evaluated using the effective electron density. Both data sets were scaled with a momentum-independent constant obtained from a $q^2$ fit to the high-momentum spectra.
• Figure 3: The QFI as a function of $q$ from IXS data on log-log scale.
• Figure 4: Result of using Kramers-Kronig on scaled IXS spectra and extracting $\epsilon(\mathbf{q)}$ below the gap ($4.5 \ eV$). The dashed lines correspond to the fit $\epsilon = 1 + e^{- \alpha q}$ for $q > 2.6$.
• Figure 5: Values of $K(\mathbf{q})$ from IXS spectra. The shaded region is defined by the bounds Eq. \ref{['eq: K Bound']} with $\epsilon(\mathbf{q})$ from IXS data used and the $\mathbf{q} \rightarrow 0$ value of the gap used. Horizontal lines correspond to the $\mathbf{q \rightarrow 0}$ limit of Eq. \ref{['eq: K Bound']} using $\epsilon$ as reported in onishiQuantumWeightFundamental2025 and the same $n$ as used to scale the data with the f-sum rule.