TPM Lecture: Z_2-invariants for topological insulators and cyclic cohomology

Who: Johannes Kellendonk (Lyon)
When: Monday, June 18, 2018 at 14:15
Where: The CP³ meeting room

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Topology plays a fundamental role in solid state physics. The most famous topological effect is the topological quantisation of the Hall conductivity: when a flat material bar is in its insulating regime then, under the influence of an external magnetic field, its transverse conductivity is quantised. Mathematically this can be explained by the fact that the transverse conductivity becomes a topological invariant in the insulating regime. The invariant is the Chern number of a (perhaps non-commutative) complex vector bundle. In recent years, materials have been proposed and experimentally investigated which show similar effects without external magnetic fields. Such materials are called topological insulators and the different topological invariants which arise when these materials are in their insulating regime characterise distinct topological phases of matter.
In particular materials with time reversal invariance (such an invariance is broken by an external magnetic field) are fascinating, as they are described by the topology of Real vector bundles or real non-commutative topology, and some of their invariants are mod-2-invariants—as if twice the conductivity is as good as no conductivity. The aim of my talk is to show how secondary pairings of cyclic cocycles with K-theory can be used to detect these invariants. This is an alternative approach to using the index pairing between K-homology and K-theory for this purpose.