Classical Dimers on Penrose Tilings [article]

Felix Flicker, Steven H. Simon, S. A. Parameswaran
2019 arXiv   pre-print
We study the classical dimer model on rhombic Penrose tilings, whose edges and vertices may be identified with those of a bipartite graph. We find that Penrose tilings do not admit perfect matchings (defect-free dimer coverings). Instead, their maximum matchings have a monomer density of 81-50φ≈ 0.098 in the thermodynamic limit, with φ=(1+√(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.
more » ... e loops connect second-nearest neighbour 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 sub-lattices, leading to a minimum monomer density of (7-4φ)/5≈ 0.106 all of one charge.
arXiv:1902.02799v2 fatcat:6shlilzqing3tl33roowzvesve