Constrained Models on Quasicrystals and Random Tilings
Some of the most important phenomena in physics arise when correlations emerge from local constraints. I will outline results for a range of constrained models in the new setting of aperiodic tilings. On the Ammann Beenker tiling we identify properties of classical dimers [1] and prove the existence of Hamiltonian cycles [2] (visiting each vertex precisely once), and thereby solve a range of problems including the three-colouring problem and the travelling salesperson problem. On the recently discovered ‘Spectre’ aperiodic monotiling we exactly solve the interacting quantum dimer model [3]. In ongoing work with Peru d'Ornellas (DIPC) we generalise these results to random graphs. Here we find that the classical problem admits a remarkable infinite-temperature confinement transition tunable by local graph structure. The quantum dimer model in periodic graphs coarse-grains to a compact U(1) gauge theory: we show that this mapping survives without periodicity, identifying the correlations of emergent particles using Kasteleyn theory.
[1] Doruk Efe Gökmen, Sounak Biswas, Sebastian D. Huber, Zohar Ringel, Felix Flicker, and Maciej Koch-Janusz. Compression theory for inhomogeneous systems. Nature Communications 15, 10214 (2024)
[2] Shobhna Singh, Jerome Lloyd, and Felix Flicker. Hamiltonian cycles on Ammann-Beenker Tilings. Physical Review X 14, 031005 (2024)
[3] Shobhna Singh and Felix Flicker. Exact solution to the quantum and classical dimer models on the spectre aperiodic monotiling. Physical Review B 109, L220303 (2024)
Zoom: https://dipc-org.zoom.us/j/93180907955
Youtube: https://youtube.com/live/15NF6GEtAII
Host
Adolfo Grushin
