Quasicrystals (QC) are a middle ground between periodic and disordered materials. They possess perfect long-range correlation, yet they are not periodic. Their study is therefore intrinsically different from periodic materials: how can we model a material that never repeats? This offers an exciting challenge for condensed matter physics, as we cannot resort to the usual toolbox provided by Bloch’s theorem. Up to now, quasicrystals have mostly been studied using either finite-sized quasicrystals or periodic approximants (where periodic boundary conditions are enforced on a large but finite patch of the quasicrystal).
Here, we propose a different and new description of optical quasicrystals: the configuration space (CS) representation. This representation – in which lattice sites are ordered in terms of shape and local surroundings - allows us to correctly account for the infinitely many possible local configurations of lattice sites.
As an example, we will detail the configuration space representation of the two-dimensional eight-fold quasicrystal (realised experimentally with ultracold atoms in our group). In this case, the CS representation maps the QC lattice onto a densely populated and compact octagon with periodic boundary conditions – which turns out to be topologically equivalent to a 2-hole torus.
Afterwards, we will show what configuration space can teach us about the Hubbard model of this 8- fold QC. Indeed, once re-expressed in configuration space, the Hubbard Hamiltonian is greatly simplified: the on-site energies, tunnelling amplitudes and on-site interaction energies become smooth and 8-fold symmetric surfaces. This configuration space picture then enables conclusions about the thermodynamic limit of the model, as well as its topological properties.
Finally, I will touch upon how the configuration space representation could be applied to Moiré quasicrystals as well as future research directions.