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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.

Abstract:

Abstract TQFTs, or anyon theories, have a large amount of addition mathematical structure which we explore in this chapter. We show that the quantum dimension which defines the weight of a loop in the diagram algebra is the same as the quantum dimension that describes the size of the many-anyon Hilbert space. We define the S-matrix in terms of diagrams, and the T-matrix in terms of twists. We show how the S- and T-matrices are related to the modular group of diffeomorphisms of the torus, and related further to the notion of central charge. We discuss how the many constraints on TQFTs enables one to develop a table of all possible sufficiently “small” TQFTs given certain very general conditions. We give tables for both modular and super-modular theories. We introduce the Kirby Color or Omega Strand as an important tool.

Abstract:

Abstract In this chapter we begin the detailed structure of anyon theories, or TQFTs. We introduce the concept of fusion and explain how it is related to the Hilbert space of the anyon systems. We emphasize the importance of locality as a principle for understanding anyons. Associativity is a strong constraint on possible particle fusion rules. As examples, we introduce both Fibonacci anyons and Ising anyons. We explain how the fusion rules of an anyon theory can be used to calculate the degeneracy of the ground state on arbitrary 2-manifolds.