the square and the triangular lattices. It was also shown there that the same ideas can be used to construct a quantum generalization of the two-dimensional classical Baxter (or eight-vertex) model. In all of these cases, the RK form of this generalized quantum dimer model has an exact ground-state wave function whose amplitudes are equal to the statistical (Gibbs) weights of an associated two-dimensional classical statistical mechanical system on the same lattice. Thus, if the classical problem happens to be a classical critical system, the associated wave function now describes a 2D problem at a quantum critical point. In Ref.  such quantum critical points were dubbed “conformal quantum critical points” since the long-distance structure of their [*ground-state wave functions*]{} exhibits 2D conformal invariance. Here we are interested in a different generalization of the QDM in which we consider dimer coverings (although not necessarily fully packed) of the square lattice. We will also consider 2D Hamiltonians whose wave functions correspond to classical interacting 2D dimer problems with local weights. Similar but independent constructions have also been proposed.[@castelnovo05; @Castelnovo06; @Poilblanc06]

Trying to be as physical and local as possible, we keep the quantum-resonance terms as simple as before (single plaquette moves), but 