Classical Dimers on Penrose Tilings

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 ≈ 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. 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 sub-lattice, 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. * flicker@physics.org † steven.simon@physics.ox.ac.uk ‡ sid.parameswaran@physics.ox.ac.uk FIG. 1. A finite section of the Penrose tiling constructed of two rhombuses (colored red and blue here).

Perpendicular space accounting of localized states in a quasicrystal

“…It would be interesting to see if this method can be combined with scaling symmetries as done for the Fibonacci chain [20] to investigate eigenstates which are not strictly localized. Similarly, other models defined on quasicrystals [17,21] which have strictly localized excitations may benefit from a perpendicular space approach.…”

“…While the system sizes in these experiments are much smaller compared to quasicrystalline solids, they are free of impurities and provide an opportunity to measure quantities which are not accessible for solids. Similarly, quasicrystalline order may lead to unexplored phases in models which are well understood for periodic systems [15][16][17]. Particularly cold atom realizations of quasicrystals may soon explore the effects of local quasicrystal structure in strongly interacting systems.…”

The Local Structure of the Fibonacci Chain and the Penrose Tiling from X-Ray Fluorescence Holography

“…In the PT, only eight different ways are possible in which tiles can meet at a vertex, which simplifies the evaluation. 21,22) Limiting the discussion to connections of the types "¸a" and "f " as examples, we can describe the important configurations (illustrated in Fig. 9): The "¸a" connection can be realized as a single "¸a" unit or via a pair of tiles (1 thick + 1 thin tile) as "a + a/¸" (red, dashed arrows).…”

Fractonic plaquette-dimer liquid beyond renormalization

“…Close-packed tiling models provide an exquisite frame-work within which to study such emergent phenomena. This line of research has been extended into several directions, including close-packed dimers on quasi-crystals [18], close-packed plaquette and cube (and also mixed [19]) models [20][21][22] on the square and cubic lattices [23]. While most close-packed tiling problems share similar behavior, including extensive(or subextensive) ground state degeneracy and zero-temperature entropy, the plaquette tiling system contains additional peculiar features as the monomer defects (unpaired sites) display restricted motion.…”