Direct observation of the Lindhard continuum using resonant inelastic X-ray scattering

Understanding the excitations of quantum materials is essential for unraveling how their microscopic constituents interact. Among these, particle-hole excitations form a particularly important class, as they govern fundamental processes such as screening, dissipation, and transport. In metals, the continuum of electron-hole excitations is described by the Lindhard function. Although central to the theory of Fermi liquids, the corresponding Lindhard continuum has remained experimentally elusive. Here, we report its direct observation in the weakly correlated metal MgB$_{2}$ using ultra-soft resonant inelastic X-ray scattering (RIXS). We resolve a linearly dispersing excitation with velocity comparable to the Fermi velocity and find quantitative agreement with simulations of the non-interacting charge susceptibility. A detailed analysis and decomposition of the simulations reveal the intra-band origin of this low-energy excitation, confirming it as the Lindhard continuum. Our results establish ultra-soft RIXS as a momentum-resolved probe of the fermiology in metals and call for deeper investigations of continuum features in RIXS and related spectroscopy of other materials beyond MgB$_{2}$.