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arXiv · 1902.02799

Classical Dimers on Penrose Tilings

Felix Flicker·Steven H. Simon·S. A. Parameswaran
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Abstract

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 $81-50φ\approx 0.098$ in the thermodynamic limit, with $φ=\left(1+\sqrt{5}\right)/2$ 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 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 $\left(7-4φ\right)/5\approx 0.106$ all of one charge.

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Felix Flicker, Steven H. Simon, S. A. Parameswaran. 2019-10-03. Classical Dimers on Penrose Tilings. https://doi.org/10.1103/physrevx.10.011005

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