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.

Figure 1

Comparison of the thermodynamic and local density fluctuations in the two-dimensional Hubbard model. Left column [(a) and (d)], $U/t=1.6(2)$; middle column [(b) and (e)], $U/t=8.2(5)$; right column [(c) and (f)], $U/t=12.0(7)$. Top row: static structure factor $S(k=0)$ measuring density fluctuations in thermodynamically large volumes. Bottom row: density fluctuations at a single site of the optical lattice. Dashed line: on-site fluctuations in the noninteracting limit. Solid line: on-site fluctuations in the infinite-interactions limit. Temperature is encoded as color.

Figure 2

Nonlocal density-density correlations for different lattice fillings. The solid line is the theoretical expectation of the ideal Fermi gas in a two-dimensional lattice. Color code: purple, $U/t=1.6(2)$; light blue, $U/t=6.1(4)$; green, $U/t=8.2(5)$; yellow, $U/t=10.3(6)$; red, $U/t=12.0(7)$.

Figure 3

Nonlocal density-density correlations for the lowest temperatures. Color code: purple, $U/t=1.6(2)$; light blue, $U/t=6.1(4)$; green, $U/t=8.2(5)$; yellow, $U/t=10.3(6)$; red, $U/t=12.0(7)$. The line shows the prediction of the ideal Fermi gas on a square lattice for $k_{B}T/t=0.65$.