Topological connections between the 2D Quantum Hall problem and the 1D quasicrystal

TL;DR

This paper establishes a direct topological connection between 2D Quantum Hall systems and 1D quasicrystals through the Fibonacci-Hall model, extending Chern number concepts and linking gap labels across dimensions.

Contribution

The paper introduces the Fibonacci-Hall model as a common ancestor that relates 2D Quantum Hall topological invariants to 1D quasicrystal gap labels, enabling a unified topological framework.

Findings

Fibonacci-Hall model links 2D and 1D topological properties.

Chern numbers are extended to 1D quasicrystal energy bands.

Gap labels in 1D are derived from 2D Chern numbers.

Abstract

1D quasicrystals such as the Fibonacci chain have been said to ``inherit" their topological properties from the 2D Quantum Hall problem. Yet, a direct way to see the connection was lacking until a common ancestor, the Fibonacci-Hall model, was introduced recently \cite{aj2025}. This 2D ancestor model relates the role of the external magnetic flux in the Hall problem and that of a geometric flux which describes the winding of the quasicrystal in 2D, in the cut-and-project method. Doing this enables us to extend the notion of Chern numbers as defined in 2D, to the energy bands of the 1D chain by adiabatic continuity. The older notion of gap labels in the 1D system are now seen to be derivable from the Chern numbers of the 2D bands. The Fibonacci-Hall model thus provides an important link between physics of two paradigmatic models, the Fibonacci quasicrystal and the quantum Hall insulator. The generalization to other 1D quasiperiodic models is expected to be relatively straightforward. The extension to 2D cut-and-project tilings is left for future studies.