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Abstract:

Abstract We explicitly work out the details of some simple examples anyon theories. We start with a fusion ring and using consistency conditions we build F-matrices, then the possible braidings (R-matrices). We consider Z2 fusion rules, Fibonacci fusion rules and Z3 fusion rules as simple cases. We consider Ising fusion rules, obtaining eight possible (modular) unitary braided theories. We then describe several more abelian theories, including the toric code, and general ZN anyons. These can be extended to describe all possible abelian braided theories in terms of a limited number of so-called “prime” theories.

Abstract:

Abstract Manifolds can be decomposed into simplicies (triangles in 2D or tetrahedra in 3D). TQFTs can be constructed as state sums over discrete quantum numbers on the on these discretized manifolds. Such sums appear like partition functions, statistical mechanics sums of Boltzmann weights. In order to for these sums to yield manifold invariants the sum must be independent on the particular simplicial decomposition (or triangulation) of the manifold. The so-called “Pachner Moves” describe all possible changes of the decomposition, so a sum which is unchanged under Pachner moves gives a manifold invariant. One such state sum is the the Turaev-Viro state sum, which takes as an input a (spherical) fusion category (F matrices satisfying the pentagon equation) and then allows one to assign a (scalar) manifold invariant to a 3D manifold. The corresponding TQFT is known as the quantum double, or Drinfeld double of the fusion category. Much of the study of the Turaev-Viro model has been in the context of so-called spin-network models of quantum gravity. A very similar state sum TQFT is the Dijkgraaf-Witten model, which (in 3D) takes as an input a group and a 3-cocyle (acting as F-matrices). The Dijkgraaf-Witten model is generalizable to any dimension.

Abstract:

Abstract We refine our notion of a topoogical phase of matter to account for symmetry protection. We describe SPT (symmetry protected topological or trivial) phases and give some history of this idea. We focus on the case of on-site symmetries. The case of one-dimensional SPTs with on-site symmetry is discussed. For understanding two-dimensional SPTs we focus on the case of Z2 symmetry and argue that there are two possible Z2 paramagnetic phases, i.e., gapped phases where the Z2 symmetry is unbroken. These two phases are in one-to-one correspondence with the two possible Z2 loop gases. We argue that with any discrete group symmetry, we generally classify SPT phases with the possible 3-cocyles. We briefly discuss the idea of symmetry defects and explain that such particles are the excitations of the corresponding twisted Kitaev quantum double model of the same cocyle. We briefly discuss the edge modes of SPT phases. Finally we argue that the SPT phases can be considered as the “ungauging” of the corresponding Dijkgraaf-Witten model.

Abstract:

Abstract Ideas similar to that of the Kauffman bracket introduced in chapter 2 can be used to generate valid anyon theories. We begin with a planar diagram algebra defined only by the value of a loop (the loop weight). One wants to build new particle types by grouping together multiple strands of loops. However, to do so one needs so-called Jones-Wenzl projectors to orthogonalize between species resulting in the so-called Temperly-Lieb Algebra. We demonstrate this construction by building a planar diagram algebra corresponding to the Z2 loop gas (bosons, fermions, semions), and then Ising Anyons. We show how this generalizes to a much bigger set of possible theories. We show how these theories can be made unitary, and how the F-matrices may be calculated. Finally we discuss how the R-matrices can be simply obtained by returning to the Kauffman bracket rules.

Abstract:

Abstract The Toric Code can also be interpreted as a phase of matter. The vertex and plaquette operators become terms of the Hamiltonian, and the ground state space becomes the code space. The various types of errors become the various types of quasiparticle excitations. We explore the braiding properties of these excitations and identify properties of the TQFT described by this phase of matter such as the S- and T-matrices. We also describe this anyon theory with a simple charge-flux model. We repeat the exercise for the ZN toric code.