Topological quasiparticles and the holographic bulk-edge relation in 2+1D string-net models

Abstract

String-net models allow us to systematically construct and classify 2+1D topologically ordered states which can have gapped boundaries. We can use a simple ideal string-net wavefunction, which is described by a set of F-matrices [or more precisely, a unitary fusion category (UFC)], to study all the universal properties of such a topological order. In this paper, we describe a finite computational method -- Q-algebra approach, that allows us to compute the non-Abelian statistics of the topological excitations [or more precisely, the unitary modular tensor category (UMTC)], from the string-net wavefunction (or the UFC). We discuss several examples, including the topological phases described by twisted gauge theory (i.e., twisted quantum double $D^α(G)$). Our result can also be viewed from an angle of holographic bulk-boundary relation. The 1+1D anomalous topological orders, that can appear as edges of 2+1D topological states, are classified by UFCs which describe the fusion of quasiparticles in 1+1D. The 1+1D anomalous edge topological order uniquely determines the 2+1D bulk topological order (which are classified by UMTC). Our method allows us to compute this bulk topological order (i.e., the UMTC) from the anomalous edge topological order (i.e., the UFC).

Figures (8)

• Figure 1: The energy density distribution of a quasiparticle.
• Figure 2: A local area with $K$ plaquettes and 4 external legs. The evaluation removes all the plaquettes.
• Figure 3: Quasiparticle $\xi$: The local energy density is constant in the ground state area but higher in the $\xi$ area.
• Figure 4: (Color online) Cut a cylinder into two cylinders. The entanglement between the two cylinders is only in the neighborhood of the cutting loop.
• Figure 5: (Color online) Gluing two cylinders: make sure there is an overlapping area between the glued boundaries (red and blue).