effects of phase separation, leading instead to a complex phase diagram of inhomogeneous phases, of which we only study its most commensurate case.

In Section \[sec:SMA\], we study the elementary quantum excitations of the doped QDMs satisfying the RK condition using the single-mode approximation. We only present the main results and have relegated the details to Appendix \[app:fks\]. We find that in the dimer-hole liquid phase, hole and dimer density fluctuations have quadratic dispersions $E(k)\sim k^2$. Thus this phase should be characterized as a Bose-Einstein condensate of bosonic charged particles (holes), but not really a superfluid, for reasons similar to those of Rokhsar and Kivelson.[@rokhsar88] We summarize our overall conclusions in Section \[sec:conclusions\].

While this manuscript was being completed (and refereed) a number of independent studies of aspects of this problem have been published.[@Castelnovo06; @Poilblanc06; @alet06] Our results agree with those in these references wherever they overlap, as noted throughout this paper.

Quantum Hamiltonians for Interacting Dimers at Finite Hole Density {#sec:hamiltonians}
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The Hamiltonian of the quantum dimer model (QDM) can be written in the Rokhsar-Kivelson (RK) form[@rokhsar88] as the sum of a set of mutually non-commuting projection operators $Q_p$, $$H=\sum_{\{p\}} Q_p \;,$$ where $\{p\}$ denotes the set of 