Condensed Matter Theory

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.