Thermodynamics versus Local Density Fluctuations in the Metal-Mott-Insulator Crossover

The crossover between a metal and a Mott insulator leads to a localization of fermions from delocalized Bloch states to localized states. We experimentally study this crossover using fermionic atoms in an optical lattice by measuring thermodynamic and local (on-site) density correlations. In the metallic phase at incommensurable filling we observe the violation of the local fluctuation-dissipation theorem indicating that the thermodynamics of the system cannot be characterized by local observables alone. In contrast, in the Mott insulator we observe the convergence of local and thermodynamic fluctuations indicating the absence of long-range density-density correlations.