On the origin of energy gaps in quasicrystalline potentials

TL;DR

This paper introduces a configuration-space framework to predict and explain energy gaps in quasicrystalline potentials, overcoming limitations of finite-size numerics and revealing a hierarchy of gaps from resonant hybridization.

Contribution

It presents a novel theoretical approach that explains the origin of energy gaps in quasicrystals and validates predictions with large-scale simulations.

Findings

Hierarchy of energy gaps from resonant hybridization

Integrated density of states pinned to irrational areas in configuration space

Excellent agreement between theory and large-scale simulations

Abstract

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.