From the Temperley-Lieb Categories to Toric Code

Abstract

In this thesis our aim is to show the ground states of Hamiltonians on 2-dimensional surfaces, which is made up by putting together stablizer operators corresponding to edges(vertices) and faces, coincide with topology of surface [Kit03]. we rst introduce the Temperley-Lieb categories and then de ne the skein module of TL( = 1), which is the specialization of the Temperley-Lieb category at = 1. Then we introduce the Levin-Wen models, which is the generalization of the toric code, with general de ned projections (operators in quantum physics) corresponding to edges and vertices. We then prove the kernel of Hamiltonian in the toric code is isomorphic to the skein module of TL( = 1) and therefore only depends on the topology of surface it bases on. Show more