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

22+3 pages, 18+4 figures

References in corpus (16)

• Non-Abelian Anyons and Topological Quantum Computation
• Local stabilizer codes in three dimensions without string logical operators
• Dirac Strings and Magnetic Monopoles in Spin Ice Dy2Ti2O7
• Fractons
• Many-Body Localization in a Quasiperiodic System
• Topological Order and Conformal Quantum Critical Points
• On the stability of U(1) spin liquids in two dimensions
• A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes
• Three dimensional resonating valence bond liquids and their excitations
• Antiferromagnetic order in the Hubbard Model on the Penrose Lattice
• Algebraic vortex liquid theory of a quantum antiferromagnet on the kagome lattice
• Noncollinear magnetic order in quasicrystals
• Non-Fermi-Liquid Behavior in Metallic Quasicrystals with Local Magnetic Moments
• Observing Monopoles in a Magnetic Analog of Ice
• Magnetism in Rare Earth Quasicrystals: RKKY Interactions and Ordering
• Superconvergence of Topological Entropy in the Symbolic Dynamics of Substitution Sequences

Cited by in corpus (19)

• Nature of Protected Zero Energy States in Penrose Quasicrystals
• Statistical physics through the lens of real-space mutual information
• Physical properties of an Aperiodic monotile: Graphene-like features, chirality and zero-modes
• Fractonic plaquette-dimer liquid beyond renormalization
• A Kitaev-type spin liquid on a quasicrystal
• Emergent Fracton Dynamics in a Non-Planar Dimer Model
• Perpendicular space accounting of localized states in a quasicrystal
• Statistical mechanics of dimers on quasiperiodic Ammann-Beenker tilings
• The frustration-free fully packed loop model
• Dulmage-Mendelsohn percolation: Geometry of maximally-packed dimer models and topologically-protected zero modes on site-diluted bipartite lattices
• Exact Solution to the Quantum and Classical Dimer Models on the Spectre Aperiodic Monotiling
• Quasicrystalline Spin Liquid
• Compression theory for inhomogeneous systems
• Ergodic Archimedean dimers
• Hamiltonian Cycles on Ammann-Beenker Tilings
• Site-selective correlations in interacting "flat-band" quasicrystals
• Ground states of a family of frustrated spin models for quasicrystals and their approximants
• Emergence of quasiperiodic behavior in transport and hybridization properties of clean lattice systems
• Tilings of a bounded region of the plane by maximal one-dimensional tiles