  We study a generalized quantum hard-core dimer model on the square and
honeycomb lattices, allowing for first and second neighbor dimers. At
generalized RK points, the exact ground states can be constructed, and
ground-state correlation functions can be equated to those of interacting 1+1
dimensional Grassmann fields. When the concentration of second neighbor dimers
is small, the ground state correlations are shown to be short-ranged
corresponding to a (gaped) spin liquid phase. On a 2-torus, the ground states
exhibit fourfold topological degeneracy. On a finite cylinder we have found a
dramatic even-odd effect depending on the circumference, and propose that this
can be used as a numerical diagnostic of gapped spin-liquid phases, more
generally.
