  A model of strongly correlated spinless fermions hopping on a checkerboard
lattice is mapped onto a quantum fully-packed loop model. We identify a large
number of fluctuationless states specific to the fermionic case. We also show
that for a class of fluctuating states, the fermionic sign problem can be
gauged away. This claim is supported by numerically evaluating the energies of
the low-lying states. Furthermore, we analyze in detail the excitations at the
Rokhsar-Kivelson point of this model thereby using the relation to the height
model and the single-mode approximation.
