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Thermodynamics vs. 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 cannot be explained by local observables. In contrast, in the Mott-insulator we observe the convergence of local and thermodynamic fluctuations indicating the absence of long--range density-density correlations.