Statistical mechanics of dimers on quasiperiodic Ammann-Beenker tilings

Abstract

We study classical dimers on two-dimensional quasiperiodic Ammann-Beenker (AB) tilings. Despite the lack of periodicity we prove that each infinite tiling admits 'perfect matchings' in which every vertex is touched by one dimer. We introduce an auxiliary 'AB${}^{\ast }$' tiling obtained from the AB tiling by deleting all 8-fold coordinated vertices. The AB${}^{\ast }$ tiling is again two-dimensional, infinite, and quasiperiodic. The AB${}^{\ast }$ tiling has a single connected component, which admits perfect matchings. We find that in all perfect matchings, dimers on the AB${}^{\ast }$ tiling lie along disjoint one-dimensional loops and ladders, separated by 'membranes', sets of edges where dimers are absent. As a result, the dimer partition function of the AB${}^{\ast }$ tiling factorizes into the product of dimer partition functions along these structures. We compute the partition function and free energy per edge on the AB${}^{\ast }$ tiling using an analytic transfer matrix approach. Returning to the AB tiling, we find that membranes in the AB${}^{\ast }$ tiling become 'pseudomembranes', sets of edges which collectively host at most one dimer. This leads to a remarkable discrete scale-invariance in the matching problem. The structure suggests that the AB tiling should exhibit highly inhomogenous and slowly decaying connected dimer correlations. Using Monte Carlo simulations, we find evidence supporting this supposition in the form of connected dimer correlations consistent with power law behaviour. Within the set of perfect matchings we find quasiperiodic analogues to the staggered and columnar phases observed in periodic systems.

arXiv comment33 pages, 26 figures

References in corpus (71)

• "Deconfined" quantum critical points
• Quantum many-body scars
• Quantum criticality beyond the Landau-Ginzburg-Wilson paradigm
• An RVB phase in the triangular lattice quantum dimer model
• Quantum Monte Carlo with Directed Loops
• Local stabilizer codes in three dimensions without string logical operators
• Spin-lattice coupling in frustrated antiferromagnets
• Quantum Glassiness
• Evidence for deconfined quantum criticality in a two-dimensional Heisenberg model with four-spin interactions
• Fractons
• Subdimensional Particle Structure of Higher Rank U(1) Spin Liquids
• Cluster Algorithm for Vertex Models
• Fracton Phases of Matter
• Short-ranged RVB physics, quantum dimer models and Ising gauge theories
• On the stability of U(1) spin liquids in two dimensions
• Fracton-Elasticity Duality
• Quantum dimer model on the kagome lattice: solvable dimer liquid and Ising gauge theory
• A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes
• Generalized Directed Loop Method for Quantum Monte Carlo Simulations
• Coulomb and Liquid Dimer Models in Three Dimensions
• Higher-Order Topological Insulators in Quasicrystals
• Glassy quantum dynamics in translation invariant fracton models
• Interacting Fibonacci anyons in a Rydberg gas
• Three dimensional resonating valence bond liquids and their excitations
• Physics of low energy singlet states of the Kagome lattice quantum Heisenberg antiferromagnet
• The number of matchings in random graphs
• Relaxation time for a dimer covering with height representation
• The phase diagram of the hexagonal lattice quantum dimer model
• Topological phases without crystalline counterparts
• Quantum Criticality and Deconfinement in Phase Transitions Between Valence Bond Solids
• Classical dimers on the triangular lattice
• An introduction to the dimer model
• The Laplacian and $\bar{\partial }$ operators on critical planar graphs
• From classical to quantum dynamics at Rokhsar-Kivelson points
• Classical dimers with aligning interactions on the square lattice
• Zero-Temperature Properties of the Quantum Dimer Model on the Triangular Lattice
• Directed Loop Updates for Quantum Lattice Models
• Antiferromagnetic order in the Hubbard Model on the Penrose Lattice
• Conventional superconductivity in quasicrystals
• Floating phase versus chiral transition in a 1D hard-boson model
• Critical eigenstates and their properties in one and two dimensional quasicrystals
• Dimers on two-dimensional lattices
• Zero-Temperature Phase Transitions of Antiferromagnetic Ising Model of General Spin on a Triangular Lattice
• Generic mixed columnar-plaquette phases in Rokhsar-Kivelson models
• Quantum dimer phases in a frustrated spin ladder: Effective field theory approach and exact diagonalization
• Pocket Monte Carlo algorithm for classical doped dimer models
• Correlations and confinement in non-planar two-dimensional dimer models
• Emergent Glassy Dynamics in a Quantum Dimer Model
• Search for quantum dimer phases and transitions in a frustrated spin ladder
• Phase diagram of the frustrated spin ladder
• Superlattice structure in the antiferromagnetically ordered state in the Hubbard model on the Ammann-Beenker tiling
• Transition to a many-body localized regime in a two-dimensional disordered quantum dimer model
• Long-range Magnetic Order in Models for Rare Earth Quasicrystals
• Vison gap in the Rokhsar-Kivelson dimer model on the triangular lattice
• Eigenstate thermalization hypothesis in quantum dimer models
• New Two-color Dimer Models with Critical Ground States
• Staggered dimer order in S=1/2 quantum spin ladder system with four spin exchange
• Magnetism in Rare Earth Quasicrystals: RKKY Interactions and Ordering
• Dual gauge theory formulation of planar quasicrystal elasticity and fractons
• Nature of Protected Zero Energy States in Penrose Quasicrystals
• Exact solution of close-packed dimers on the kagome lattice
• Classical Dimers on Penrose Tilings
• DMRG investigation of constrained models: from quantum dimer and quantum loop ladders to hard-boson and Fibonacci anyon chains
• Graphene with vacancies: supernumerary zero modes
• Maximum matching on random graphs
• Dulmage-Mendelsohn percolation: Geometry of maximally-packed dimer models and topologically-protected zero modes on site-diluted bipartite lattices
• A mean-field monomer-dimer model with random monomer activities. Exact solution and rigorous results
• Emergence of cooperative dynamics in fully packed classical dimers
• Exact Solution and Correlations of a Quantum Dimer Model on the Checkerboard Lattice
• Dimer Models, Free Fermions and Super Quantum Mechanics
• Quasicrystals

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• Compression theory for inhomogeneous systems
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• Ground states of a family of frustrated spin models for quasicrystals and their approximants