We analyze the quantum ground state structure of a specific model
of itinerant, strongly interacting lattice fermions. The interactions are
tuned to make the model supersymmetric. Due to this, quantum ground
states are in one-to-one correspondence with cohomology classes of the
so-called independence complex of the lattice. Our main result is a complete
description of the cohomology, and thereby of the quantum ground
states, for a two-dimensional square lattice with periodic boundary conditions.
Our work builds on results by Jonsson, who determined the
Euler characteristic (Witten index) via a correspondence with rhombus
tilings of the plane. We prove a theorem, first conjectured by Fendley,
which relates dimensions of the cohomology at grade $n$ to the number of
rhombus tilings with $n$ rhombi.