of the superposition of these specially weighted configurations. The ground-state wave function, $\ket{G}$, the state annihilated by all the projection operators, for this system is: $$\ket{ G}=\frac{1}{\sqrt{Z(w^2)}}\sum_{\{\mathcal{C}\}} w^{\displaystyle{N_p[\mathcal{C}]}} \ket{\mathcal{C}} \;,
\label{Gzw}$$ where $Z(w^2)$, the normalization of this state, $$\begin{aligned}
 Z(w^2) =\sum_{\{\mathcal{C}\}} w^{\displaystyle{2N_p[\mathcal{C}] } }
 \label{z1}\end{aligned}$$ has the form the partition function of classically interacting dimers with a coupling $u=-2\ln w$ between parallel neighboring dimers. In the following, we will assume an attractive coupling, $u<0$ or $w>1$. The case $u>0$ was studied for the fully-packed case in Ref. .

There are two different ways in which we can add doping to our system, while still being able to determine the ground state. If we add the following fine-tuned hole-related terms to the initial Hamiltonian  $$\begin{aligned}
  H^{\rm hole}_{\rm canonical}= - t_{\rm hole}\sum_{<ijk>}\Bigg\{\Bigg[\Big|\raisebox{-0.5ex}{\includegraphics{Ham22.eps}}\Big>\Big<\raisebox{-0.5ex}{\includegraphics{Ham23.eps}}\Big| + \mathit{h.c.}\Bigg]  \nonumber\\
  -\Big|\raisebox{-0.5ex}{\includegraphics{Ham22.eps}}\Big>\Big<\raisebox{-0.5ex}{\includegraphics{Ham22.eps}}\Big| -\Big|\raisebox{-0.5ex}{\includegraphics{Ham23.eps}}\Big>\Big<\raisebox{-0.5ex}{\includegraphics{Ham23.eps}}\Big|\Bigg\}
\label{Ham2}\end{aligned}$$ then the resulting ground state becomes $$\ket{ G_{N_h}}=\frac{1}{\sqrt{Z(w^2)}}\sum_{\{\mathcal{C}_{N_h}\}}
  w^{\displaystyle{N_p[\mathcal{C}_{N_h}]}}\ket{\mathcal{C}_{N_h}} \;,
  \label{Gzw2}$$ where the number of holes $N_h$ is now fixed at a specified value. The norm of this wave-function, $Z(w^2)$, is the *canonical* partition function for the set of dimer coverings with a fixed number of holes.

On the other hand, if we add the following terms, which do not conserve the number of 