Winding Numbers and Topology of Aperiodic Tilings

TL;DR

This paper demonstrates that the diffraction properties of 1D quasicrystals can be understood through a topological invariant, the Čech cohomology group, which encodes combinatorial information of aperiodic tilings.

Contribution

It introduces a constructive method to compute the Čech cohomology for various aperiodic tilings and relates it to diffraction features via winding numbers.

Findings

Čech cohomology encodes all relevant combinatorial information.

Winding numbers relate topological invariants to diffraction features.

Topological features show resilience against perturbations.

Abstract

We show that diffraction features of $1D$ quasicrystals can be retrieved from a single topological quantity, the \v{C}ech cohomology group, $\check{H}^{1}\cong\mathbb{Z}^2$, which encodes all relevant combinatorial information of tilings. We present a constructive way to calculate $\check{H}^{1}$ for a large variety of aperiodic tilings. By means of two winding numbers, we compare the diffraction features contained in $\check{H}^{1}$ to the gap labeling theorem, another topological tool used to label spectral gaps in the integrated density of states. In the light of this topological description, we discuss similarities and differences between families of aperiodic tilings, and the resilience of topological features against perturbations.