Exact results for strongly-correlated fermions in 2+1 dimensions

We derive exact results for a model of strongly-interacting spinless fermions hopping on a two-dimensional lattice. By exploiting supersymmetry, we find the number and type of ground states exactly. Exploring various lattices and limits, we show how the ground states can be frustrated, quantum critical, or combine frustration with a Wigner crystal. We show that on generic lattices, the model is in an exotic ``super-frustrated'' state characterized by an extensive ground-state entropy.