Topological Equivalence between the Fibonacci Quasicrystal and the Harper Model

TL;DR

This paper demonstrates that the Fibonacci quasicrystal and the Harper model are topologically equivalent through a continuous deformation that preserves bulk gaps, unifying their boundary phenomena and explaining experimental observations.

Contribution

It introduces a general model showing the topological equivalence between Fibonacci quasicrystals and the Harper model, regardless of modulation type.

Findings

Models are topologically equivalent without closing bulk gaps.

Boundary phenomena are shared between the models.

Equivalence extends to any Fibonacci-like quasicrystal.

Abstract

One-dimensional quasiperiodic systems, such as the Harper model and the Fibonacci quasicrystal, have long been the focus of extensive theoretical and experimental research. Recently, the Harper model was found to be topologically nontrivial. Here, we derive a general model that embodies a continuous deformation between these seemingly unrelated models. We show that this deformation does not close any bulk gaps, and thus prove that these models are in fact topologically equivalent. Remarkably, they are equivalent regardless of whether the quasiperiodicity appears as an on-site or hopping modulation. This proves that these different models share the same boundary phenomena and explains past measurements. We generalize this equivalence to any Fibonacci-like quasicrystal, i.e., a cut and project in any irrational angle.