Non Commutative Geometry of Tilings and Gap Labelling

TL;DR

This paper develops a non-commutative geometric framework for tilings, constructing associated C*-algebras and computing their K-theory to understand spectral gap labeling in quantum models.

Contribution

It introduces a non-commutative space and algebra for tilings, computes the K_0-group for one-dimensional cases, and links gap labels to invariant measures in substitution tilings.

Findings

K_0-group computed for 1D tilings and products

Gap labels determined by invariant measures

Integrated density of states fully describes gap labels

Abstract

To a given tiling a non commutative space and the corresponding C*-algebra are constructed. This includes the definition of a topology on the groupoid induced by translations of the tiling. The algebra is also the algebra of observables for discrete models of one or many particle systems on the tiling or its periodic identification. Its scaled ordered K_0-group furnishes the gap labelling of Schroedinger operators. The group is computed for one dimensional tilings and Cartesian products thereof. Its image under a state is investigated for tilings which are invariant under a substitution. Part of this image is given by an invariant measure on the hull of the tiling which is determined. The results from the Cartesian products of one dimensional tilings point out that the gap labelling by means of the values of the integrated density of states is already fully determined by this measure.