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 $81-50\varphi\approx 0.098$ in the thermodynamic limit, with $\varphi=\left(1+\sqrt{5}\right)/2$ 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 $\left(7-4\varphi\right)/5\approx 0.106$ all of one charge.
Highlights
Dimer models are convenient abstractions of the physics of energetic constraints arising from strong correlations
Turning briefly to the wider class of Penrose-like tilings, we prove that a variation on the Penrose tiling made instead from tiles shaped as darts and kites is unable to admit perfect matchings
III, we prove that Penrose tilings do not admit perfect matchings, i.e., they must feature a finite density of monomer defects, and study properties of the boundaries which restrict the movement of monomers
Summary
Dimer models are convenient abstractions of the physics of energetic constraints arising from strong correlations. Studying dimer coverings of graphs remains an active area of current research in mathematics and statistical physics [18,19,20,21,22,23] For both the classical and quantum cases, results to date have focused primarily on periodic graphs, partly because of their relative simplicity and the resulting potential for exact results and partly because of the relevance to physical systems such as crystal lattices [11,19]. We prove that certain other examples cannot admit perfect matchings We demonstrate that these latter cases feature broadly similar behavior to the rhombic Penrose tiling.
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