Aperiodic Topological Boundary Modes: Revisiting Quasi-Periodic Localization

Although physical systems are generically aperiodic, most work on non-interacting topological materials focuses on translation-invariant cases. These systems are well described by band theory and perturbatively resistant to the disorder of real world conditions, however, the regime of strong aperiodicity is not frequently discussed in the context of topological materials. Aiming to better understand such systems, this work generalizes key results in translation-invariant systems to their aperiodic counterparts. We then apply these techniques to the canonical Andre-Aubrey-Harper (AAH) Model and its 1D Metal-Insulator Transition (MIT). We uncover a deep connection between the known non-commutative topological properties of the model and the MIT. This not only highlights the power of this non-commutative technology, but also separates the AAH model from its disordered counterparts. The 1D MIT in the AAH model and its peculiar properties are indeed topological in nature. This suggests quasiperiodic systems form their own special class of topological materials distinct from their translation-invariant counterparts.