typically disordered and are topological fluids.[@moessner01a]

The more physically relevant *doped* QDMs, with a finite density of charge carriers (holes), are much less understood, although some properties are known.[@fradkin90a; @balents05b; @alet05; @syljuasen05; @alet06; @Castelnovo06; @Poilblanc06] In QDMs a spin-$\frac{1}{2}$ hole fractionalizes into a bosonic [*holon*]{}, an excitation that carries charge but no spin, and a [*spinon*]{}, a fermionic excitation that carries spin but no charge.[@kivelson87; @rokhsar88; @fradkin90a; @fradkin91] Holons can be regarded as sites that do not belong to any dimer, whereas spinon pairs are broken dimers. This form of electron fractionalization is observable in the spectrum of these systems only in the topological disordered (spin-liquid) phases of the undoped QDM. Otherwise, as in the case of the valence bond crystalline states which exhibit long range dimer order, spinons and holons are [*confined*]{} and do not exist as independent excitations.[@fradkin91]

In this paper, we consider several interacting QDMs on a square lattice at finite hole doping, and discuss their possible phases and phase transitions as a function of hole density and strength of the interactions. At any finite amount of doping the system will have a finite density of holes, which are hard core charged bosons in this description. To 