| Topological phases are new kind of quantum phases of matter with properties robust against weak disorders and interactions. They occur in two-dimensional electron liquids with quantized Hall conductance and in topological insulators etc. The description of these phases goes beyond Landau's theory of symmetry breaking. They are (partially) characterized by exotic properties, such as topology-dependent ground state degeneracy(GSD), fractional quantum numbers of anyonic excitations and topology-protected bulk-edge duality etc.;In this dissertation, we systematically examine exactly solvable discrete models, particularly the so-called Levin-Wen models, for two-dimensional topological phases. They were expected to describe a large class of nonchiral (or, time reversal invariant) two-dimensional topological phases and to provide a Hamiltonian approach to some topological quantum field theories, which are related to topological invariants defined in the mathematical literature. We first show how to construct concrete models of the Levin-Wen type on a two-dimensional graph (generalized lattice), associated with the data from representation theory (the 3j- and 6j-symbols) of finite groups or quantum groups. Then an operator approach is developed to deal with the properties of the models, such as topology-dependent GSD and fractional quantum numbers for quasiparticle excitations. In this approach we are able to demonstrate the topological invariance/symmetry of the models under the mutation transformations of the graph on which the system lives, and explore this invariance to compute the topology-dependent GSD on a torus. Moreover, we use the operator approach to study the fluxon excitations, i.e., quasiparticles living on plaquettes, and to exhibit their fractional exchange (braiding) and exclusion statistics. Also, we explicitly show the correspondence between the degenerate ground states and the quasiparticle excitations: (1) the GSD on a torus is equal to the number of quasiparticle species; and (2) the modular matrices S and T obtained from the modular transformation of the torus for the ground states coincide with those obtained from the fractional exchange statistics of quasiparticles. In this way the present study reveals the first time in the literature the Hilbert space structure for the degenerate ground states as well as that for the excited states, and the interconnection between them in the Levin-Wen models. |
