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Topological Properties of Matter
David Carpentier, Laboratoire de Physique de l'ENS Lyon
Topology is a branch of mathematics which focuses on properties of objects insensitive to smooth
deformations. Its first occurrence in the description of matter happened in the 70s : while ordered
phases are identified by their broken symmetries, topological numbers allow to classify stable
defects of these ordered phases, such as vortices, dislocations or skyrmions. With the discovery
of the Quantum Hall Effect in 1980, it was soon realized that a quantum electronic phase as a
whole could be associated with a topological property. In this presentation, I will first describe these
pioneering works which were awarded the 2016 Nobel prize. Then I will turn to the more recent
evolution of this topological characterization of matter. This includes the discovery of insulators of
a new kind, which are identified not by a standard broken symmetry but by a topological property
of their ground state. More recently, topological semi-metals were identified : metals in which
electrons behave like relativistic particles, and which constitute analogs in electronic phases of
topological defects.