hole density.

Interacting Dimers at Zero Hole Density {#sec:zero-density}
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It is a well known fact that both classical and quantum two-dimensional dimer models can be represented in terms of height models. For the classical case, this mapping is well known[@blote82a; @blote82b; @henley97a; @ardonne04]. The mapping for the quantum case has also been discussed extensively[@fradkin90a; @levitov90; @zheng-sachdev89; @fradkin91; @ardonne04; @fradkin04]. In both cases the mapping relates each dimer configuration to a configuration of a set of integer-valued (height) variables, $h(\vec
r)$, which reside on the sites of the dual lattice, a square lattice in the case of interest here. Thus, this mapping amounts to a duality transformation.

An alternative picture follows from realizing that dimer configurations can be mapped onto the degenerate ground-state configurations of fully frustrated Ising models[@blote82a; @moessner03a]. In our case, the corresponding spin model is the fully frustrated Ising model on the square lattice (FFSI) at zero temperature. In the Ising model picture, each dimer is dual to an unsatisfied bond of the fully frustrated Ising model and classical dimer interactions correspond to second neighbor interactions in the square lattice FFSI[@fradkin81]. It is easy to see that holes correspond to unfrustrated plaquettes in the FFSI. In this 