Constrained Models on Quasicrystals

Some of the most important phenomena in condensed matter physics, such as fractionalisation and topological order arise when strong correlations emerge from local constraints. Examples include dimer models (tiling a chess board with dominoes), emergent magnetic monopoles in the spin ice materials, and resonating valence bond solids. We outline results for a range of constrained models in a new setting: aperiodic long-range ordered Ammann Beenker tilings (AB), which have the symmetries of certain known quasicrystals. Treating the vertices and edges of AB as those of bipartite graphs, we (i) prove the existence of Hamiltonian cycles (visiting each vertex precisely once), an NP-complete problem in general graphs, and thereby construct polynomial-time solutions in AB to a range of NP-complete problems with applications in adsorption, catalysis, scanning tunneling microscopy, and elsewhere; (ii) demonstrate the existence of fully-packed loops on Ammann Beenker tilings, and study their statistics numerically, giving a generalisation of ice-type models to aperiodic settings; (iii) apply MPS-DMRG to study the quantum dimer model; we are able to apply this technique in 2D owing to structures emerging naturally from the dimer constraint.