Statistical mechanics of dimers on quasiperiodic Ammann-Beenker tilings

TL;DR

This paper analyzes classical dimer models on quasiperiodic Ammann-Beenker tilings, proving the existence of perfect matchings, computing partition functions, and revealing scale-invariance and complex correlation behaviors.

Contribution

It introduces a novel approach to studying dimers on quasiperiodic tilings, including the AB$^*$ tiling, and uncovers scale-invariance and inhomogeneous correlations in the matching problem.

Findings

Existence of perfect matchings on quasiperiodic tilings.

Partition function factorizes along disjoint structures.

Evidence of power-law decay in dimer correlations.

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$^*$' tiling obtained from the AB tiling by deleting all 8-fold coordinated vertices. The AB$^*$ tiling is again two-dimensional, infinite, and quasiperiodic. The AB$^*$ tiling has a single connected component, which admits perfect matchings. We find that in all perfect matchings, dimers on the AB$^*$ 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$^*$ 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$^*$ tiling using an analytic transfer matrix approach. Returning to the AB tiling, we find that membranes in the AB$^*$ 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.