Theoretical Generalizations of Topological Phases and Topological Entanglement Entropy

In this thesis, we explore topological phases, study their properties, and present some novel extensions of them. Our study of topological phases begins on simplified theoretical models and provides a tractable setting to discuss the otherwise abstract mathematics involved, while maintaining the power to capture many experimentally salient features of these phases. We generalize these theoretical models and discover new phases of matter, topological flux phases, which are topological phases with a uniform anyonic flux. Other extensions of topological phases we study are fractionalized Fermi liquids, which are gapless topological phases with non-trivial interactions between gapless and topological sectors. Finally, we also focus on one peculiar property of topological phases, topologi- cal entanglement entropy, which captures the fact that some information is distributed globally in topological phase and can only be accessed with a topologically non-trivial measurement