August 22, 2013
10:00am (334 JFB)
Title: "Emergent properties in two-dimensional exactly solvable models for topological phases"
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 thesis, 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 non-chiral (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 3/j/- and 6/j/-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 quasi-particle 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 particle species of quasiparticles: (1) the GSD on a torus is equal to the number of particle species of quasiparticles; and (2) the modular matrices /S/ and /T/ obtained from the modular transformation of the torus for the ground states are equal to 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 for the excited states, and the interconnection between them, in the Levin-Wen models.