Correlations and Collective Modes in Fermions on Lattices

Experiments with ultracold atoms present an outstanding opportunity for implementing novel tests of theories of strongly correlated fermions. We present three new theoretical methods to treat these systems. Luttinger liquid theory and the exact one-dimensional solutions let us calculate collective mode frequencies at the metal-insulator transition in a lattice. A new Gaussian phase-space method for fermion systems can be used to simulate finite temperature atomic correlations. Finally, we introduce an approximate diagrammatic technique which correctly includes molecule-molecule interactions, and gives an accurate, quantitative theory of the BEC-BCS crossover regime.