Origin of energy gaps in quasicrystalline potentials

Quasicrystals, structures that are ordered yet aperiodic, defy conventional band theory and have long resisted analytical predictions, confining most studies to finite-size real-space numerics. In particular, it is not obvious that quasicrystalline potentials should support true energy gaps: such gaps are a hallmark of periodic systems, but are generally absent in fully disordered materials. Here, we overcome this limitation with a configuration-space framework that predicts and explains the positions and microscopic origins of energy gaps in an experimentally realizable eightfold optical quasicrystal. We show that a hierarchy of true gaps arises from resonant hybridization between increasingly distant neighboring sites, and that the integrated density of states below these gaps is pinned 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 work establishes a route to analytical understanding of quasicrystalline potentials in the infinite-size limit and provides a rigorous foundation for future studies of their transport and many-body quantum properties.