Some of the most important phenomena in physics arise when correlations emerge from local constraints. Examples include dimer models (tiling a chess board with dominoes) and 'magnetic monopole' excitations in crystals called spin ices. We outline results for a range of constrained models in a new setting: aperiodic Ammann Beenker tilings (AB). These lesser-known cousings of Penrose tilings have the symmetries of certain exotic materials called quasicrystals. We prove the existence of Hamiltonian cycles (visiting each vertex precisely once), and thereby solve a range of related problems including the three-colouring problem and the travelling salesperson problem [1]. Potential applications include adsorption, scanning tunneling microscopy, and protein folding. Using machine learning (RSMI-NE) we identify an emergent discrete scale invariance to the structure of dimer matchings [2]. In ongoing work we apply the density matrix renormalisation group (DMRG) to the quantum dimer model.
Irene Rosana Giardina