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 tilings, with the symmetries of certain exotic materials called quasicrystals.
On the 'Ammann Beenker tiling' 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. On the recently discovered 'Spectre' aperiodic monotiling we provide an exact analytic solution to the interacting quantum dimer model [2]. The result features deconfined particle-like excitations at all interaction strengths, which is impossible in the periodic square and hexagonal tilings.
[1] Shobhna Singh, Jerome Lloyd, and Felix Flicker, “Hamiltonian cycles on Ammann-Beenker Tilings”, arXiv:2302.01940
[2] Shobhna Singh and Felix Flicker, “Exact Solution to the Quantum and Classical Dimer Models on the Spectre Aperiodic Monotiling”, arXiv:2309.14447
