Quantitative determination of temperature in the approach to magnetic order of ultracold fermions in an optical lattice.

We perform a quantitative simulation of the repulsive Fermi-Hubbard model using an ultracold gas trapped in an optical lattice. The entropy of the system is determined by comparing accurate measurements of the equilibrium double occupancy with theoretical calculations over a wide range of parameters. We demonstrate the applicability of both high-temperature series and dynamical mean-field theory to obtain quantitative agreement with the experimental data. The reliability of the entropy determination is confirmed by a comprehensive analysis of all systematic errors. In the center of the Mott insulating cloud we obtain an entropy per atom as low as 0.77k(B) which is about twice as large as the entropy at the Néel transition. The corresponding temperature depends on the atom number and for small fillings reaches values on the order of the tunneling energy.