Quasicrystals are long-range ordered but not periodic, so they present an attractive challenge to condensed matter physics because they cannot rely on the usual toolbox based on Bloch’s theorem. We present a theorem-free numerical method for constructing the Hubbard-Hamiltonian of aperiodic potentials and apply it to the case of recently realized 2D optical quasicrystals with 8-fold rotational symmetry using cold atoms. To do. We construct maximally localized Wannier functions and use them to extract on-site energies, tunnel amplitudes, and interaction energies. Furthermore, it introduces a site-ordered configuration-space representation in terms of shape and local environment, leading to a compact description of infinite-sized quasicrystals where all Hamiltonian parameters can be expressed as smooth functions. This configuration-space diagram makes it possible to construct arbitrarily large tight-binding graphs for numerical many-body computations, enabling new analytical discussions about the topological structure and many-body physics of these models. For example, the conclusion that this quasicrystal hosts the unit. Fill the Mott insulator at the thermodynamic limit.
