  The exchange coupling between quantum mechanical spins lies at the origin of
quantum magnetism. We report on the observation of nearest-neighbor magnetic
spin correlations emerging in the many-body state of a thermalized Fermi gas in
an optical lattice. The key to obtaining short-range magnetic order is a local
redistribution of entropy within the lattice structure. This is achieved in a
tunable-geometry optical lattice, which also enables the detection of the
magnetic correlations. We load a low-temperature two-component Fermi gas with
repulsive interactions into either a dimerized or an anisotropic simple cubic
lattice. For both systems the correlations manifest as an excess number of
singlets as compared to triplets consisting of two atoms with opposite spins.
For the anisotropic lattice, we determine the transverse spin correlator from
the singlet-triplet imbalance and observe antiferromagnetic correlations along
one spatial axis. Our work paves the way for addressing open problems in
quantum magnetism using ultracold fermions in optical lattices as quantum
simulators.
