holes in the system, to the Hamiltonian  $$\begin{aligned}
  H^{\rm hole}_{\rm {grand-canonical}} =
  - \tilde t_{\rm hole}\sum_{links}\Bigg\{ \Bigg[\Big|\raisebox{-0.5ex}{\includegraphics[scale=0.6]{Ham20.eps}}\Big>\Big<\raisebox{-0.5ex}{\includegraphics[scale=0.6]{Ham21.eps}}\Big| + h.c.\Bigg]  \nonumber\\
  - z^2\Big|\raisebox{-0.5ex}{\includegraphics[scale=0.6]{Ham20.eps}}\Big>\Big<\raisebox{-0.5ex}{\includegraphics[scale=0.6]{Ham20.eps}}\Big|-z^{-2}\Big|\raisebox{-0.5ex}{\includegraphics[scale=0.6]{Ham21.eps}}\Big>\Big<\raisebox{-0.5ex}{\includegraphics[scale=0.6]{Ham21.eps}}\Big| \Bigg\}
  \nonumber \\
  &&
\label{Ham3}\end{aligned}$$ then the ground state of the system becomes $$\ket{ G}=\frac{1}{\sqrt{Z(w^2,z^2)}}\sum_{\{\mathcal{C}\}}
  w^{\displaystyle{N_p[\mathcal{C}]}} z^{\displaystyle{N_h[\mathcal{C}]}}
  \ket{\mathcal{C}} \;,
\label{Gzw3}$$ with $$Z(w^2,z^2)=\sum_{\{\mathcal{C}\}} w^{2N_p[\mathcal{C}]}
 z^{2N_h[\mathcal{C}]} \;.
 \label{Zzw}$$ Equation  includes a natural, short-range repulsion between neighboring holes and an off-diagonal term which represents creation-annihilation of dimers. Furthermore, there is a dimer fugacity term, as in every perturbative derivation of a quantum dimer model.[@rokhsar88] We note that Eq.  has the same form as a grand partition function for dimer coverings of the square lattice. This partition function now depends not only on the interaction $u$ defined below Eq. , but also on the hole chemical potential $\mu/|u| = 2\ln z$.

Since the canonical and grand-canonical ensembles become equivalent in the thermodynamic limit, the two ground-state wave functions and  must correspond to the same ground-state physics. Furthermore, it is clear that for $w=1$ the models are located at the usual RK point of the quantum dimer model on the square lattice. For a system with periodic boundary conditions, each configuration $\mathcal{C}$ contains only an *even* number of holes, with half of the 