Classical Dimers on Penrose Tilings

We study the classical dimer model on rhombic Penrose tilings, whose edges and vertices may be identified as those of a bipartite graph. We find that Penrose filings do not admit perfect matchings (defect-free dimer coverings). Instead, their maximum matchings have a monomer density of 81 - 50 phi approximate to 0.098 in the thermodynamic limit, with phi = (1 + root 5)/2 the golden ratio. Maximum matchings divide the tiling into a fractal of nested closed regions bounded by loops that cannot be crossed by monomers. These loops connect second-nearest-neighbor even-valence vertices, each of which lies on such a loop. Assigning a charge to each monomer with a sign fixed by its bipartite sublattice, we find that each bounded region has an excess of one charge, and a corresponding set of monomers, with adjacent regions having opposite net charge. The infinite tiling is charge neutral. We devise a simple algorithm for generating maximum matchings and demonstrate that maximum matchings form a connected manifold under local monomer-dimer rearrangements. We show that dart-kite Penrose tilings feature an imbalance of charge between bipartite sublattices, leading to a minimum monomer density of (7 - 4 phi)/5 approximate to 0.106 all of one charge.