The subtle interplay between kinetic energy, interactions, and dimensionality challenges our comprehension of strongly correlated physics observed, for example, in the solid state. In this quest, the Hubbard model has emerged as a conceptually simple, yet rich model describing such physics. Here we present an experimental determination of the equation of state of the repulsive two-dimensional Hubbard model over a broad range of interactions 0≲U/t≲20 and temperatures, down to k_{B}T/t=0.63(2) using high-resolution imaging of ultracold fermionic atoms in optical lattices. We show density profiles, compressibilities, and double occupancies over the whole doping range, and, hence, our results constitute benchmarks for state-of-the-art theoretical approaches.

We experimentally study the emergence of antiferromagnetic correlations between ultracold fermionic atoms in a two-dimensional optical lattice with decreasing temperature. We determine the uniform magnetic susceptibility of the two-dimensional Hubbard model from simultaneous measurements of the in situ density distribution of both spin components. At half filling and strong interactions our data approach the Heisenberg model of localized spins with antiferromagnetic correlations. Moreover, we observe a fast decay of magnetic correlations when doping the system away from half filling.

We measure entropy and short-range correlations of ultracold fermionic atoms in an optical lattice for a range of interaction strengths, temperatures, and fillings. In particular, we extract the mutual information between a single lattice site and the rest of the system from a comparison between the reduced density matrix of a single lattice site and the thermodynamic entropy. Moreover, we determine the single-particle density matrix between nearest neighbors from thermodynamic observables and show that even in a strongly interacting Mott insulator fermions are significantly delocalized over short distances in the lattice.

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.

We coherently manipulate spin correlations in a two-component atomic Fermi gas loaded into an optical lattice using spatially and time-resolved Ramsey spectroscopy combined with high-resolution in situ imaging. This novel technique allows us not only to imprint spin patterns but also to probe the static magnetic structure factor at arbitrary wave vector, in particular the staggered structure factor. From a measurement along the diagonal of the 1 st Brillouin zone of the optical lattice, we determine the magnetic correlation length and the individual spatial spin correlators. At half filling, the staggered magnetic structure factor serves as a sensitive thermometer for the spin temperature, which we employ to study the thermalization of spin and density degrees of freedom during a slow quench of the lattice depth.