  Using perturbative expansions and the contractor renormalization (CORE)
algorithm, we obtain effective hard-core bosonic Hamiltonians describing the
low-energy physics of $S=1/2$ spin-dimer antiferromagnets known to display
supersolid phases under an applied magnetic field. The resulting effective
models are investigated by means of mean-field analysis and quantum Monte Carlo
simulations. A "leapfrog mechanism", through means of which extra singlets
delocalize in a checkerboard-solid environment via correlated hoppings, is
unveiled that accounts for the supersolid behavior.
