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Abstract:

Abstract The arguments about the toric code relied on diagrammatic reasoning about loops around handles of the torus, but the underlying lattice was not crucial to any of the arguments. We now decsribe the same model more abstractly, with diagrams in the continuum, with no reference to the underlying lattice at all. We find that we are using a planar diagram algebra (in this case a d=1, Z2 loop gas) and the toric code ground states arise as a sum over all possible such diagrams. We introduce the notion of the tube algebra. We build so-called “idempotent” states on an annulus (or “tube”) and from these we can extract quasiparticle types, the S-matrix, the T-matrix, braiding, and fusion properties. We then generalize the construction to the ZN toric code.

Abstract:

Abstract We give a basic introduction to the Aharonov-Bohm effect, and use this effect to construct a toy-model of anyons known as the“charge-flux” model. We explore some of the basic properties of this model, including fusion of anyons, antiparticles, and the degeneracy of the ground state on a torus, which is related to the number of species of anyons. This degeneracy can in principle serve as a quantum memory.

Abstract:

Abstract A mechanism for constructing one anyon theory from another is anyon condensation, akin to Bose condensation. It is believed that any continuous phase transition between TQFTs must occur by such a condensation transition. We describe the main steps in such a condensation: Identification/Splitting, and Confinement. We give simple examples when the condensing boson is a simple current then describe the more general structure. We explain how the idea of a Chern-Simons coset can be understood under the framework of condensation. Finally we discuss the relationship between condensation and gappable boundaries.

Abstract:

Abstract A theory with anyon permuting symmetry is unchanged under permutation of the names of the particle types. The toric code is an example of a theory with such a symmetry. The toric code, and all theories with such symmetries can be viewed as arising as a condensation of anyons from some more complicated parent theory. Given that such a symmetry exists, we consider defects of this symmetry which generalize the mathematical structure of the TQFT to a so-called G-crossed extension. To clarify how this works we use a particularly tranparent example of the toric code where the symmetry of particle types is reduced to a lattice symmetry, and the defects correspond to dislocations in the lattice. It can be shown that these defects harbor Majorana zero modes. We discuss the general structure of a parent theory condensing via a G-crossed theory to a theory with permutation symmetry. Finally we mention theories with both on-site symmetries and anyons, which are known as symmetry enriched topological phases.