Jean-Noël Fuchs (LPTMC)

In this talk, I will try to give a pedagogical overview of phases of matter that are classified using ideas from topology rather than symmetry. I will insist on the difference between two families of such phases of matter: topological insulators and topological order.

On the one hand, topological insulators can be seen as a refinement of band insulators, in which bands carry topological invariants (for example, Chern numbers) such that several classes of insulators can be distinguished. This is a generalization of ideas of D. Thouless and co-workers concerning the integer quantum Hall effect. Interactions do not play a major role in this context.

On the other hand, topological order is a notion proposed by X.-G. Wen that takes its origin in the fractional quantum Hall effect, which crucially depends on interactions between electrons. A topologically-ordered system is characterized by a robust ground-state degeneracy that depends on the genus of the surface on which the system is placed (whether a sphere or a torus, for example) and it features fractionalized quasiparticles known as anyons. Important aspects are long-range quantum entanglement and the non-local nature of the order.