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Topological states of matter and noncommutative geometry

Bourne, Christopher

Description

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...[Show more]

dc.contributor.authorBourne, Christopher
dc.date.accessioned2015-12-02T01:11:29Z
dc.date.available2015-12-02T01:11:29Z
dc.identifier.otherb37881309
dc.identifier.urihttp://hdl.handle.net/1885/16960
dc.description.abstractThis 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.
dc.language.isoen
dc.subjectNoncommutative geometry
dc.subjectKK-theory
dc.subjecttopological insulators
dc.subjecttopological phases
dc.titleTopological states of matter and noncommutative geometry
dc.typeThesis (PhD)
local.contributor.supervisorCarey, Alan
local.contributor.supervisorcontactalan.carey@anu.edu.au
dcterms.valid2015
local.type.degreeDoctor of Philosophy (PhD)
dc.date.issued2015-08
local.identifier.doi10.25911/5d6cfd6408e6d
local.mintdoimint
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