The paper studies the interactions between dynamical systems, represented via groupoid operator algebras, and index theory, described using Kasparov's bivariant K-functor. The key motivating example is the dynamical systems associated to Delone sets.

Delone sets give a model of aperiodic atomic configurations, where the material may be amorphous or quasi-crystalline. The atomic configuration does not have a group structure, but key properties can be encoded in a discrete groupoid. We construct bivariant K-cycles, a generalisation of K-theory classes, that encode the dynamics of the aperiodic lattice. For the case of quasi-crystalline configurations, we obtain finer information coming from the long-range aperiodic order of the material.

We then use our constructed K-cycles to define bulk topological phases of aperiodic materials. A bulk-boundary correspondence is also shown, which arises from factorisation properties of the constructed K-cycles.

Chris Bourne, Bram Mesland
"Index theory and topological phases of aperiodic lattices"
Journal Reference: Ann. Henri Poincare 20 (6) (2019), 1969--2038
doi: 10.1007/s00023-019-00764-9
arXiv: 1807.03972