holes on either sublattice.

We remark that the fact that the exact ground-state wave function is a sum (as opposed to a product) of the ground states of sectors labeled by the number of holes on the lattice, is due to the resonance term that we have used to represent the motion of holes. In particular, we have assumed that a dimer can *break* into two holes which themselves repel each other. In the limit of very strong hole-hole repulsions, in strong-coupling perturbation theory, it is straightforward to recover a fixed-hole density sector with a single-hole resonance move involving three sites in any direction; the coupling strength in Eq.  then becomes $t_{\rm hole}\sim z^4\tilde t_{\rm hole}$, and thus, in the limit $z\rightarrow 0$, it reduces to an effective hopping amplitude for the holes.

Mean-Field Results {#sec:mean-field}
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To examine the physics described by the ground-state wave functions obtained in the previous section we start with a discussion of a mean-field theory of the phase diagram. We use the standard approach of regarding the probability densities, obtained by squaring the wave function, as the Gibbs weights of a classical two-dimensional system and focus on the interacting dimer model on the square 