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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.

Abstract:

Abstract Given a discrete group, we consider planar diagrams where each edge is labeled with an element of the group, and vertices must obey the rule that multiplication of the edges incident on the vertex (in order) must yield the identity. The possible F-matrices for such a diagram are discussed and are known as 3-cocycles from the formalism of group cohomology. We discuss a simple example of building anyons from the ZN group. Noncommutative groups can be considered although they cannot describe a braided theory.

Abstract:

Abstract A number of tools for determining properties of TQFTs are described. In particular we outline the use of the computer program “Kac”. We also give pointers to other useful resources.