simplify the problem, in this work we do not consider the physical effects of the charge-neutral fermionic spinons which in principle should also be present. Thus, at this level of approximation, all spin-carrying excitations are effectively projected out. The remaining degrees of freedom are thus dimers (“spin-singlet valence bonds”) and charged hard-core bosonic holes. Already this simplified picture of a strongly correlated system is very non-trivial.

For a certain relation between its coupling constants, known as the Rokhsar-Kivelson (RK) condition,[@rokhsar88] QDM Hamiltonians, both with and without holes, can be written as a sum of projection operators. These RK Hamiltonians are manifestly positive definite operators. For this choice of couplings, the ground-state wave function is a zero-energy state which is known exactly. This [*RK wave function*]{} is a local function of the degrees of freedom of the QDM, the local dimer and hole densities. The quantum-mechanical amplitudes of the RK wave functions turn out to have the same form as the Gibbs weights of a two-dimensional (2D) *classical* dimer problem with a finite density of holes. For the generalized doped QDMs that we consider, the norm of the exact ground-state wave function is equal to the partition function of a system 