  The antiferromagnetic Ising model on a checkerboard lattice has an ice-like
ground state manifold with extensive degeneracy. and, to leading order in J_xy,
deconfined spinon excitations. We explore the role of cyclic exchange arising
at order J^2_xy/J_z on the ice states and their associated spinon excitations.
By mapping the original problem onto an equivalent quantum six--vertex model,
we identify three different phases as a function of the chemical potential for
flippable plaquettes - a phase with long range Neel order and confined spinon
excitations, a non-magnetic state of resonating square plaquettes, and a
quasi-collinear phase with gapped but deconfined spinon excitations. The
relevance of the results to the square--lattice quantum dimer model is also
discussed.
