Electrons in most everyday objects barely interact with one another. But in some forms of quantum matter, such as high-temperature superconductors, these interactions cannot be ignored. The microscopic physics of these materials, known as “strongly correlated matter,” is very difficult to understand theoretically. The leading theoretical model, known as the Hubbard model, can be solved for simple systems but is much harder to deal with for complex ensembles of particles. Here, we use ultracold atoms to create an experimental realization of the Hubbard model in two dimensions and, using a novel approach, determine microscopic correlations among the particles from precise thermodynamic measurements.

In our experiment, we study ensembles of potassium isotopes (${}^{40}\mathrm{K}$) in the two lowest hyperfine states, which are stored in an optical lattice. We investigate the transition from a metallic phase to a Mott insulator (an insulator that, according to traditional band theory, should be a conductor) by probing the density distribution inside the lattice and determining the pressure, entropy, and kinetic energy for a range of interactions, temperatures, and fillings. The kinetic energy is proportional to the first-order correlation function between neighboring lattice sites and therefore measures the degree of localization of the atoms. We also demonstrate the development of correlations between a lattice site and its environment upon a reduction of temperature.

Our technique provides access to the full thermodynamic entropy without the need for more complex measurements, and it can be used to evaluate new cooling schemes for strongly correlated quantum gases.