Bragg spectrum, K-theory and Gap Labelling of aperiodic solids

TL;DR

This paper explores the relationship between the diffraction spectrum, topological invariants, and gap labels in aperiodic solids, linking physical spectral properties to mathematical topological invariants.

Contribution

It establishes a connection between the topological Bragg spectrum, Chern numbers, and the gap-labelling group in aperiodic solids.

Findings

Topological Bragg spectrum relates to eigenvalues of the dynamical system.

Chern numbers are connected to the topological invariants of the solid.

The gap-labelling group describes possible spectral gaps in the Schrödinger operator.

Abstract

The diffraction spectrum of an aperiodic solid is related to the group of eigenvalues of the dynamical system associated with the solid. Those eigenvalues with continuous eigenfunctions constitute the topological Bragg spectrum. We relate the topological Bragg spectrum to the topological invariants (Chern numbers) of the solid and to the gap-labelling group, which is the group of possible gap labels for the spectrum of a Schr\"odinger operator describing the electronic motion in the solid.