Abstract
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 tilings do not admit perfect matchings (defect-free dimer coverings). Instead, their maximum matchings have a monomer density of in the thermodynamic limit, with 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 all of one charge.
15 More- Received 25 February 2019
- Revised 3 October 2019
DOI:https://doi.org/10.1103/PhysRevX.10.011005
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Fill a chess board with dominoes, remove one, and then shift and rotate the remaining dominoes so that the two exposed black-and-white squares appear to move around the board. This simple game—an example of a classical “dimer model”—provides a powerful analogy for studying particle-antiparticle pairs, charge fractionalization, and even high-temperature superconductors. Here, we apply this model to Penrose tilings, aperiodic mosaics built from rhombus-shaped tiles.
We ask whether it is possible to color the edges of the tiles such that each vertex connects to precisely one colored edge. The answer is no, and we find the minimum number of uncovered vertices that must exist. This number has a connection to the famous golden ratio, a number that appears repeatedly in classical architecture and nature.
The fact that the minimum number of uncovered vertices is nonzero is interesting. Just as the domino analogy shows that dimer models can represent particles being created from the vacuum, for example, this finite density shows that the theory has particles present in the vacuum itself. Even stranger is that there are particles of opposite charge simultaneously present in the vacuum. They are unable to annihilate with one another because of the underlying constraints on dimer placement implied by the structure of the Penrose tiling.
Our analysis takes the dimer model to a new setting, beyond the paradigm of periodicity and disorder. This in turn might lead to new insights into theories featuring topological order, including topological quantum computation and high-temperature superconductivity.