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Abstract:

Abstract Having established the structure of the Hilbert space of multiple anyons, we now discuss the effects of braiding the anyons around each other. We introduce the idea of an R-matrix and emphasize the importance of locality. Using R and F together we can fully describe any braiding of identical particles. We again use Fibonacci and Ising anyons as simple examples.

Abstract:

Abstract We give a brief account of some of the experimental systems believed to harbor anyons (or that might someday be coaxed to harbor anyons). We discuss Fractional Quantum Hall Effects — abelian and nonabelian — in some detail. We mention Fractional Chern Insulators, and Bosonic Fractional Quantum Hall Effects. We discuss gapped spin liquids — particular the Kitaev Honeycomb model and frustrated antiferromagents. We briefly discuss conventional superconductors before turning to so-called Majorana materials. Finally we discuss the recent advances in quantum simulation of matter.

Abstract:

Abstract We give a brief discussion of some things that have been left out of the book. In particular this includes detailed theory of fractional quantum Hall effect, connection to conformal field theory, tensor networks, and topological insulators.

Abstract:

Abstract We have constructed diagrammatic rules for evaluating labeled knots and links (and graphs) embedded in S3. However, a 3D TQFT should be able to give us information about any 3-manifold. In order to describe arbitrary 3-manifolds we use Dehn surgery — a manifold is represented as a link embedded in S3. To obtain the manifold, the strands of the link are thickened to a solid torus which is excised from the manifold and replaced with longitude and meridian exchanged. The Lickorish-Wallace theorem assures us that any closed orientable 3D manifold can be obtained in this way. The link representation a 3-manifold is not unique — links related by a series of Kirby moves represent the same manifold. The Witten-Reshitikhin-Turaev invariant is a link invariant which remains unchanged by Kirby moves and therefore corresponds to a manifold invariant of the represented 3-manifold. This invariant is the formal construction of the Chern-Simons manifold invariant which does not require reference to ill-defined concepts like functional integrals. The surgery approach also gives a nice proof (using a construction known as Chain-Mail) of the Turaev-Walker-Roberts theorem that the Turaev-Viro invariant for a Chern-Simons theory is the absolute square of the Chern-Simons (or Witten-Reshitikhin-Turaev) invariant.