Some of the most complex and vexing issues in electronic materials are modeled by extremely simple Hamiltonians.  High-temperature superconductors, for example, may arise from magnetic interactions in a Mott insulating state, described by the simple Hubbard model.  The Hubbard model stipulates that particles (electrons in the case of superconductors) are distributed in a square lattice where they can hop from site to site with a tunneling energy t, and where they may interact with occupied nearest neighbor sites with interaction energy U.  No one knows whether this simple “hydrogen-atom” model actually gives rise to the d-wave pairing underlying the cuprate superconductors.

I will describe two experiments that use ultracold atoms in an optical lattice as stand-ins for the electrons in ionic lattices:  1) the Hubbard model in 3D; and 2) the polarized spin-½ Fermi gas in 1D.  In the first experiment, we are searching for the anti-ferromagnetic Mott insulating state that is expected to exist above the superconducting transition when there is exactly one-atom per lattice site.  We have used Bragg scattering of near-resonant light to characterize the lattice, and will use a spin-sensitive variant of this tool to detect magnetic correlations.  In the second experiment, we have used an optical lattice in two-dimensions to create a bundle of 1D tubes containing an imbalanced two spin-state mixture of ⁶Li fermions.  The phase diagram of this system contains three phases:  a fully-paired superfluid, a fully-polarized ferromagnet, and a partially polarized state that is predicted to be the exotic FFLO superfluid state, for which the pairs have non-zero center of mass momentum.