  We study a model of strongly correlated S=1/2 fermions on the planar
pyrochlore, or checkerboard, lattice, at fractional (1/8) filling. Starting
with the extended Hubbard model in the limit of strong Coulomb repulsion,
low-energy configurations can be mapped onto hard-core dimer configurations
whose dimers carry a spin degree of freedom. An effective Hamiltonian similar
to the kinetic term of a quantum dimer model on the square lattice which
rotates two parallel dimers (in a spin-singlet configuration) by 90 degrees
naturally emerges. We also introduce an additional term in the Hamiltonian, a
generalized dimer plaquette interaction, in order to realize a closer analogy
to the latter model. For a strong dimer plaquette attraction stabilizing a
columnar phase, a spontaneous dimerization takes place in the direction of the
columns of (spin-carrying) dimers. Using exact diagonalizations of
two-dimensional periodic clusters, the analysis of the low-energy spectrum and
of several types of correlation functions gives indeed evidence for a new type
of lattice symmetry breaking phase, the eight-fold degenerate Mixed
Columnar-Plaquette Crystal, and for a transition from this phase to a
Resonating-Singlet-Pair Crystal (found in previous studies) which restores the
rotational symmetry of the lattice. Similar conclusions and phase diagram are
also reached from a simple variational approach.
