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Abstract:

Abstract One generalization of the toric code is the so-called Kitaev Quantum Double model which realizes the Drinfeld double of a group. In this model edges of a lattice are directed and labeled with elements of a group G which then give us a planar diagram algebra. We consider mainly the “untwisted” case where the F-matrices (or equivalently 3-cocyles) are trivial. This generalizes the toric code where the group is Z2. Again we have vertex and plaquette operators, and a code space which satisfies both. As with the toric code, the ground state is topological although here (depending on the group G) the quasiparticles may be nonabelian. We turn to a continuum model for a more detailed analysis of the ground state space and the quasiparticles using the tube algebra, which, using some amount of group theory, is easily tractable. We discuss the so-called quasiparticle “ribbon operators” which generalize the error strings of the toric code. We relate the Kitaev quantum double model to lattice gauge theory and briefly discuss its generalization to higher dimensions.

Abstract:

Abstract The Levin-Wen model is a general construction that uses an input planar diagram algebra to construct a Hamiltonian with a topologically ordered ground state (the Drinfeld double of the input category). The ground state (code) space consists of a sum over all diagrams weighted by the evaluation of the diagram. We work through the example of the doubled Fibonacci model. Here the input category is the Fibonacci fusion rules and F-matrices. We work with continuum diagrams and construct the full tube algebra of the model. We identify the quasiparticle types to include both left- and right-handed Fibonacci anyons. In the case that the input category admits a modular braiding, much of the analyis simplifies and the output is two copies of the input category of opposite chiralities (both left- and right-handed). We finally turn to the construction of the explicit Hamiltonian on the lattice, and construction of quasiparticle string (or ribbon) operators. We return to the reconsider the toric code and doubled semion models as simple examples.

Abstract:

Abstract Many of the theories that one runs into most often are particularly simple in that they have a higher degree of symmetry than a generic TQFT or anyon theory needs to have. In this chapter we construct a simplified set of rules for evaluating diagrams for such simpler theories.

Abstract:

Abstract We introduce the idea of the path integral and use it to study the properties of identical particles under exhange. We explain why the topology of paths through space-time, the fundamental group of the configuration space (the braid group in 2+1 dimensions and the permutation group in 3+1 dimensions) determines the possible particle types. We argue that in 3+1 dimensions one only has bosons and fermions but in 2+1 dimensions other particles, known as anyons, can exist. We describe both the abelian and nonabelian type of anyon and argue why nonabelian anyons could be useful for quantum computation. We introduce the notion of the “quantum dimension” of an anyon which describes the size of the many anyon Hilbert space.

Abstract:

Abstract We return to study planar diagram algebras as in chapters 8 and 9, but now more formally. We define diagrams as being operators that can be "stacked" on top of each other. We pay particular attention to properties of diagrams such as completeness of states and orthonormality of basis vectors, and we again review the use of F-matrices for change of basis. We discuss the implications of causality for these diagrams, and to what extent diagrams can be deformed in the plane. We give explicit detailed rules for evaluating planar diagrams.