Hubbard models for quasicrystalline potentials

Quasicrystals are long-range ordered, yet not periodic, and thereby present a fascinating challenge for condensed matter physics, as one cannot resort to the usual toolbox based on Bloch's theorem. Here, we present a numerical method for constructing the Hubbard Hamiltonian of nonperiodic potentials without making use of Bloch's theorem and apply it to the case of an eightfold rotationally symmetric two-dimensional optical quasicrystal that was recently realized using cold atoms. We construct maximally localized Wannier functions and use them to extract onsite energies, tunneling amplitudes, and interaction energies. In addition, we introduce a configuration-space representation, where sites are ordered in terms of shape and local environment, that leads to a compact description of the infinite-size quasicrystal in which all Hamiltonian parameters can be expressed as smooth functions. The configuration-space picture serves as an aperiodic analog of the Brillouin zone, and allows one to efficiently describe the quasicrystal in the thermodynamic limit, enabling new analytic arguments on the topological structure and many-body physics of these models. For instance, we use it to conclude that this quasicrystal will host unit-filling Mott insulators in the thermodynamic limit.