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.

33 pages, 26 figures

References in corpus (25)

• "Deconfined" quantum critical points
• Quantum criticality beyond the Landau-Ginzburg-Wilson paradigm
• Local stabilizer codes in three dimensions without string logical operators
• Evidence for deconfined quantum criticality in a two-dimensional Heisenberg model with four-spin interactions
• 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
• Generalized Directed Loop Method for Quantum Monte Carlo Simulations
• Three dimensional resonating valence bond liquids and their excitations
• An introduction to the dimer model
• Classical dimers with aligning interactions on the square lattice
• Antiferromagnetic order in the Hubbard Model on the Penrose Lattice
• Conventional superconductivity in quasicrystals
• Critical eigenstates and their properties in one and two dimensional quasicrystals
• Correlations and confinement in non-planar two-dimensional dimer models
• Search for quantum dimer phases and transitions in a frustrated spin ladder
• Superlattice structure in the antiferromagnetically ordered state in the Hubbard model on the Ammann-Beenker tiling
• Eigenstate thermalization hypothesis in quantum dimer models
• 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
• Maximum matching on random graphs
• Dulmage-Mendelsohn percolation: Geometry of maximally-packed dimer models and topologically-protected zero modes on site-diluted bipartite lattices
• Dimer Models, Free Fermions and Super Quantum Mechanics
• Quasicrystals

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