On the origin of energy gaps in quasicrystalline potentials

Quasicrystals, structures that are ordered yet aperiodic, defy conventional band theory, confining most studies to finite-size real-space numerics. We overcome this limitation with a configuration-space framework that predicts and explains the positions and origins of energy gaps in quasicrystalline potentials. We find that a hierarchy of gaps stems from resonant hybridization between increasingly distant neighboring sites, pinning the integrated density of states below these gaps to specific irrational areas in configuration space. Large-scale simulations of a lowest-band tight-binding model built from localized Wannier functions show excellent agreement with these predictions. By moving beyond finite-size numerics, this study advances the understanding of quasicrystalline potentials, paving the way for new explorations of their quantum properties in the infinite-size limit.