Discrete scale invariant fixed point in a quasiperiodic classical dimer model

Abstract: We study close-packed dimers on the quasiperiodic Ammann-Beenker (AB) graph, that was recently shown to have the unusual feature that hard-core dimer constraints are exactly reproduced at successive discrete length scales. This observation led to a conjecture that it would be possible to construct an exact real-space decimation scheme where each iteration preserves both the quasiperiodic tiling structure and the constraint. Here, we confirm this conjecture by explicitly constructing the corresponding renormalization group transformation and show, using large-scale Monte Carlo simulations, that the dimer distributions flow to a fixed point with non-zero dimer potentials. We use the fixed-point Hamiltonian to demonstrate the existence of slowly decaying dimer correlations. We thus identify a remarkable example of a classical statistical mechanical model whose properties are controlled by the fixed point of an exact renormalization group procedure exhibiting discrete scale invariance but lacking translational and continuous rotational symmetries.