all plaquettes of the square lattice. Each projection operator $Q_p$ acts on the states of the dimers and holes of a plaquette $p$ (or set of plaquettes surrounding $p$). In the simplest case[@rokhsar88] each $Q_p$ acts only on the states labeled by the dimer occupation numbers of the links of the plaquette $p$. In this case, the ground state is described by the short-range RVB wave function[@kivelson87] $$\ket{\textrm{RVB}}=\sum_{ \{ \mathcal{C} \} } \ket{\mathcal{C}} \;,$$ where $\{ \mathcal{C} \} $ is the set of (fully packed) dimer coverings of the 2D square lattice, and $\{ \ket{\mathcal{C}} \}$ is a complete set of orthonormal states. If one regards the dimers as spin singlet states (with the spins residing on the lattice sites) each configuration represents a set of spin singlets or valence-bonds.[@anderson87; @kivelson87] The dimer representation ignores the over-completeness of the valence bond singlet states.[@rokhsar88] This problem can be made parametrically small using a number of schemes, including large $N$ approximations[@read-sachdev89] and decorated generalizations of the spin-$1/2$ Hamiltonians.[@Raman05]

It is possible and straightforward to generalize the QDM construction so as to include other types of interactions and coverings. In Ref.  it was shown how to extend this structure to smoothly interpolate between 