Statistical mechanics of dimers on quasiperiodic Ammann-Beenker tilings

We study classical dimers on two-dimensional quasiperiodic Ammann-Beenker (AB) tilings. Using the discrete scale-symmetry of quasiperiodic tilings, we prove that each infinite tiling admits “perfect matchings”, where every vertex is touched by one dimer. We show the appearance of so-called monomer pseudomembranes. These are sets of edges, which collectively host exactly one dimer, which bound certain eightfold-symmetric regions of the tiling. Regions bounded by pseudomembranes are matched together in a way that resembles perfect matchings of the tiling itself. These structures emerge at all scales, suggesting the preservation of collective dimer fluctuations over long distances. We provide numerical evidence, via Monte Carlo simulations, of dimer correlations consistent with power laws over a hierarchy of different lengthscales. We also find evidence of rich monomer correlations, with monomers displaying a pattern of attraction and repulsion to different regions within pseudomembranes, along with signatures of deconfinement within certain annular regions of the tiling.