A non-commutative framework for topological insulators (Q2806157)

One motivation for the authors writing the article under review is to give some background on the application of Kasparov theory to topological states of matter. There have been some important contributions from mathematical physicists in understanding the mechanics of insulator systems. The purpose of the paper under review is to highlight what is understood and what remains open. This paper is an investigation of the links between topological states of matter and real Kasparov theory.NEWLINENEWLINETopological insulators are materials that are electrical insulators in the bulk but can conduct electricity on their surface via special surface electronic states. The underlying cause is time-reversal symmetry: their physics is independent of whether time is flowing backward or forward. In topological insulators, the spin-orbit interaction is so strong that the insulating energy gap is inverted -- the states that should have been at high energy above the gap appear below the gap. This twist in the order of electronic states, cannot be ``unwound''. The unusual properties of these materials has generated a lot of interest in the condensed-matter community in recent years. Topological insulators can be loosely described as physical systems possessing certain symmetries which give rise to invariants topologically protected by these symmetries.NEWLINENEWLINEIn the paper under review, the authors show that Kasparov theory in the real case may be used to give a noncommutative framework for topological insulators in which torsion invariants may be detected and effectively computed. Invariants considered in this paper are (real) Kasparov classes, and so automatically topological invariants. These Kasparov modules can be naturally identified with Clifford modules. This procedure is in contrast to the more usual methods of obtaining \(Z_2\)-invariants in the literature.