Applications of the Pfaffian State to Topological Phases

Fractional topological insulators are electronic topological phases in $(3+1)$ dimensions enriched by time reversal and charge $U(1)$ conservation symmetries. The most straightforward series of fermionic fractional topological insulators is analyzed where their bulk quasiparticles consist of deconfined partons that carry fractional electric charges in integral units of $e^\ast=e/(2n+1)$ and couple to a discrete $\mathbb{Z}_{2n+1}$ gauge theory. This thesis proposes massive symmetry preserving or breaking fractional topological insulator surface states. By combining the long-ranged entangled bulk with these topological surface states, the novel topological order of quasi-$(2+1)$ dimensional fractional topological insulator slabs, as well as their corresponding edge conformal field theories, are deduced.

Weyl and Dirac semi-metals in three dimensions have robust gapless electronic band structures. Symmetries such as lattice translation, (screw) rotation, and time reversal protect the massless single-body energy spectra. This thesis discusses many-body interactions in these systems. Here the focus is on strong interactions that preserve symmetries and are outside the single-body mean-field regime. Mapping a Dirac semi-metal to a model based on a three-dimensional array of coupled Dirac wires shows two things: (1) The Dirac semi-metal can acquire a many-body excitation energy gap without breaking the relevant symmetries, and (2) interaction can enable an anomalous Weyl semi-metallic phase that is otherwise forbidden by symmetries in the single-body setting and can only be present holographically on the boundary of a four-dimensional weak topological insulator. Both of these topological states support fractional gapped (gapless) bulk (respective boundary) quasiparticle excitations.