  We consider fermionic fully-packed loop and quantum dimer models which serve
as effective low-energy models for strongly correlated fermions on a
checkerboard lattice at half and quarter filling, respectively. We identify a
large number of fluctuationless states specific to each case, due to the
fermionic statistics. We discuss the symmetries and conserved quantities of the
system and show that for a class of fluctuating states in the half-filling
case, the fermionic sign problem can be gauged away. This claim is supported by
numerical evaluation of the low-lying states and can be understood by means of
an algebraic construction. The elimination of the sign problem then allows us
to analyze excitations at the Rokhsar-Kivelson point of the models using the
relation to the height model and its excitations, within the single-mode
approximation. We then discuss a mapping to a U(1) lattice gauge theory which
relates the considered low-energy model to the compact quantum electrodynamics
in 2+1 dimensions. Furthermore, we point out consequences and open questions in
the light of these results.
