  Cold atom systems provide a rich platform to realize strongly interacting
condensed matter systems, and recent progress in fluorescence imaging technique
has enabled identification of nontrivial doublon, singlon, and holon
correlation functions. We show that these correlators can be used to identify
the conditions under which local moments form in an interacting electronic
system. Toward this end, we report a Determinantal Quantum Monte Carlo (DQMC)
study of such correlation functions in the two-dimensional repulsive Fermi
Hubbard model on a square lattice as a function of doping, interaction strength
and temperature. We find definite signatures of the crossover from small U(band
regime) to large U(correlated insulator regime). Our key findings are: (1) An
opening of a charge gap in the thermodynamic density of states is accompanied
by the appearance of temperature insensitive points in the equation of state at
finite doping, which can be used to distinguish the band regime in cold atom
experiments. (2) Nearest neighbor doublon holon correlations track the opening
of charge gap; these compete with density correlations to generate moment
moment correlations that show different behavior in the metallic and correlated
insulator regime. (3) Non local correlation functions can be used to
distinguish between the two regimes, both at and away from half filling. Our
results allow comparisons of different correlation functions with recent
experimental findings and guide further experimental investigations.
