We define a quantum monomer-dimer model on a Penrose tiling (a quasicrystal) in the space of maximal dimer coverings. Monomers are necessarily present because it was shown by F. Flicker et al., PRX 10, 011005 (2020) that there are no perfect dimer coverings of Penrose tilings. Despite the presence of a finite density of monomers, our model has a Rokhsar-Kivelson (RK) point at which the ground state is a uniform superposition of all maximal dimer coverings. We map the model to a Z2 gauge theory with matter and calculate the dimer-dimer, vison-vison and Fredenhagen-Marcu order parameters to characterize the phase of the system using classical Monte Carlo at the RK point. We find that dimer-dimer and the vison-vison correlators decay exponentially with the distance. The Fredenhagen-Marcu order parameters corresponding to the monomers and the visons are found to approach a nonzero constant as the loops are made bigger, indicating that the gauge theory is in the confined phase.

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