Felix Flicker

University of Bristol

The Physics of Aperiodic Monotiles

Can there exist a single shape that tiles the plane, but only without translational symmetry? This question likely dates back at least 800 years, and was considered by Penrose, Hilbert, and Kepler. In March 2023 an affirmative answer was provided by Dave Smith -- a retired print technician with no formal mathematical training -- in the form of a range of 'aperiodic monotiles' such as 'The Hat' and 'The Spectre'.

I will outline recent results in which we introduce physical models to these aperiodic monotiles. First, we study electrons in a magnetic field on The Hat tiling. The spectral function shows intriguing similarities to graphene, albeit with aperiodic modifications. We find a finite density of electronic zero modes which strictly localise when the magnetic field is tuned precisely to half a flux quantum per tile. Second, we introduce 'dimer models' -- in which edges (dimers) are constrained not to meet -- to the Spectre tiling. Originally introduced to model molecular adsorption, Rokhsar and Kivelson added quantum superpositions of dimers to capture the physics of high-temperature superconductors. Remarkably, on the Spectre tiling both quantum and classical dimer models admit exact solutions, even in the presence of interactions. We find deconfined particle-like excitations at all interaction strengths, which is impossible in the periodic square and hexagonal tilings.

[1] Justin Schirmann, Selma Franca, Felix Flicker, Adolfo G. Grushin,
"Physical properties of the Hat aperiodic monotile: Graphene-like
features, chirality and zero-modes", arXiv:2307.11054

[2] Shobhna Singh and Felix Flicker, "Exact Solution to the Quantum and
Classical Dimer Models on the Spectre Aperiodic Monotiling",
arXiv:2309.14447