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A non-commutative framework for topological insulators

Journal Article


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Abstract


  • We study topological insulators, regarded as physical systems giving rise to topological invariants determined by symmetries both linear and anti-linear. Our perspective is that of non-commutative index theory of operator algebras. In particular, we formulate the index problems using Kasparov theory, both complex and real. We show that the periodic table of topological insulators and superconductors can be realized as a real or complex index pairing of a Kasparov module capturing internal symmetries of the Hamiltonian with a spectral triple encoding the geometry of the sample’s (possibly non-commutative) Brillouin zone.

Publication Date


  • 2016

Citation


  • Bourne, C., Carey, A. L. & Rennie, A. (2016). A non-commutative framework for topological insulators. Reviews in Mathematical Physics: a journal for survey and expository articles in the field of mathematical physics, 28 (2), 1650004-1-1650004-51.

Scopus Eid


  • 2-s2.0-84962672775

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=6774&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/5746

Start Page


  • 1650004-1

End Page


  • 1650004-51

Volume


  • 28

Issue


  • 2

Abstract


  • We study topological insulators, regarded as physical systems giving rise to topological invariants determined by symmetries both linear and anti-linear. Our perspective is that of non-commutative index theory of operator algebras. In particular, we formulate the index problems using Kasparov theory, both complex and real. We show that the periodic table of topological insulators and superconductors can be realized as a real or complex index pairing of a Kasparov module capturing internal symmetries of the Hamiltonian with a spectral triple encoding the geometry of the sample’s (possibly non-commutative) Brillouin zone.

Publication Date


  • 2016

Citation


  • Bourne, C., Carey, A. L. & Rennie, A. (2016). A non-commutative framework for topological insulators. Reviews in Mathematical Physics: a journal for survey and expository articles in the field of mathematical physics, 28 (2), 1650004-1-1650004-51.

Scopus Eid


  • 2-s2.0-84962672775

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=6774&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/5746

Start Page


  • 1650004-1

End Page


  • 1650004-51

Volume


  • 28

Issue


  • 2