Doped singlet-pair crystal in the Hubbard model on the checkerboard lattice

In the limit of large nearest-neighbor and on-site Coulomb repulsions, the Hubbard model on the planar pyrochlore lattice maps, near quarter filling, onto a doped, quantum, fully packed loop model. The phase diagram exhibits at quarter filling a quantum state of matter, the resonating singlet-pair crystal, an insulating phase breaking lattice symmetry. Properties of a few doped holes or electrons are investigated. In contrast to the doped quantum antiferromagnet, phase separation is restricted to very small hopping, leaving an extended ``window'' for superconducting pairing. However, the latter is more fragile for large hopping than in the case of the antiferromagnet. The existence of a fermionic supersolid is discussed.