  The highly frustrated Heisenberg antiferromagnet on Checkerboard and
Pyrochlore lattices is subject to strong quantum fluctuations. This problem is
amenable to the Contractor Renormalization (CORE) algorithm, which
systematically computes the effective interactions in a basis of local cluster
eigenstates. CORE was recently used to derive the Plaquette Boson-Fermion model
for the square lattice Hubbard model (cond-mat/0108087). For the spin half
Checkerboard and Pyrochlore cases, we find singlet ground states which break
lattice symmetry, and a gap to spin one excitations. The low excitations are
singlets, coupled by Ising like interactions at an emergent low energy scale,
which for the Pyrochlore is about 1% of the Heisenberg exchange. We discuss low
temperature thermodynamics and new interpretations of finite size numerical
data. We argue that our results are common to many models of quantum
frustration.
