The Physics of Aperiodic Monotiles

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• Felix Flicker, University of Bristol
• Thursday 23 November 2023, 12:30-13:45
• Theory Library.

If you have a question about this talk, please contact Dr Hannah Price.

**Note earlier start time**

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

This talk is part of the Theoretical Physics Seminars series.

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