Topological states of matter and noncommutative geometry

Abstract

This thesis examines topological states of matter from the perspective of noncommutative geometry and KK-theory. Examples of such topological states of matter include the quantum Hall e ect and topological insulators. For the quantum Hall e ect, we consider a continuous model and show that the Hall conductance can be expressed in terms of the index pairing of the Fermi projection of a disordered Hamiltonian with a spectral triple encoding the geometry of the sample's momentum space. The presence of a magnetic eld means that noncommutative algebras and methods must be employed. Higher dimensional analogues of the quantum Hall system are also considered, where the index pairing produces the `higher-dimensional Chern numbers' in the continuous setting. Next we consider a discrete quantum Hall system with an edge. We show that topological properties of observables concentrated at the boundary can be linked to invariants from a boundary-free model via the Kasparov product. Hence we obtain the bulk-edge correspondence of the quantum Hall e ect in the language of KK-theory. Finally we consider topological insulators, which come from imposing (possibly anti-linear) symmetries on condensed-matter systems and studying the invariants that are protected by these symmetries. We show how symmetry data can be linked to classes in real or complex KK-theory. Finally we prove the bulk-edge correspondence for topological insulator systems by linking bulk and edge systems using the Kasparov product in KKO-theory.