Dimer models on quasicrystals

How many ways are there of tiling a chess board with dominoes? This is an example of a classical dimer model. Removing one domino reveals the two squares it covered: one black, one white. These can be thought of as a particle-antiparticle pair, with further domino re-arrangements allowing the particles to move around on the board. This simple model turns out to have deep implications for the theory of strongly-interacting systems.

This project will consider dimer models on quasicrystals: aperiodic tilings which can be constructed as slices through higher-dimensional crystals. Can dominoes (dimers) be placed on the tiling's edges such that every vertex connects to one dimer? Or must there be a finite density of particle-like defects not connected to dimers? The advantage of working with quasicrystals is that they are random enough to get non-trivial results, but ordered enough that these results can be proven analytically. In previous work [Phys Rev X 10, 011005 (2020)] we found the minimum density of defects for such models in Penrose tilings, the original quasicrystals discovered by Prof. Sir Roger Penrose (recipient of the 2020 Nobel Prize for Physics).

This project has a broad range of possible routes depending on interest. These include extending to quantum dimer models such as the Rokhsar-Kivelson model (originally introduced to model high-temperature superconductivity, this model features other exotic states such as quantum spin liquids); disorder effects; confinement and the mass gap (one of the Clay Institute's Millennium Prize Problems); related models such as loop-dimer models and Hamiltonian cycles. This project can be taken in a numerical direction (using classical and quantum Monte Carlo techniques and machine learning), or an analytical direction, seeking exact results. There is scope to work with experimental groups in Cardiff and elsewhere, for example implementing similar models in mesoscopic magnetic arrays.

The project will be central to a range of work being carried out in collaboration with theorists and numericists around the globe, including close collaboration with groups in Oxford and Cornell.