Interplay of Symmetry and Interactions in Topological Phases of Matter

The interplay of symmetry and many-body interactions in electronic states can lead to the emergence of fractional point-like and loop-like excitations. These excitations have fractional charge and spin degrees of freedom and are known as topological order. In this work, we present a study of realizing three-dimensional topological order in condensed matter systems. We study symmetries and many-body interactions in three models, (1) Dirac (semi)metal, (2) Dirac nodal superconductor and (3) anomalous interacting Weyl (semi)metal, which is otherwise forbidden in the single-body setting.
Weyl and Dirac (semi)metals in three dimensions have robust gapless electronic band structures. Their massless single-body energy spectra are protected by symme- tries such as lattice translation, (screw) rotation, and time reversal. In this thesis, I will discuss many-body interactions in these systems. I will focus on strong interactions that preserve symmetries and are outside the single-body mean-field regime. By mapping a Dirac (semi)metal to a model based on a three-dimensional array of coupled Dirac wires, I will show that the Dirac (semi)metal can acquire a many-body excitation energy gap without breaking the relevant symmetries, and interaction can enable an anomalous Weyl (semi)metallic phase that is otherwise forbidden by symmetries in the single-body setting.
I will then extend this model to the superconducting analog of Dirac (semi)metals. These topological nodal superconductors possess gapless low energy excitations that are characterized by a point or line nodal Fermi surfaces. Using a coupled wire construction, I will study topological nodal superconductors that have protected Dirac nodal points. Within this model, we demonstrate many-body interactions that preserve the underlying symmetries and introduce a finite excitation energy gap. These gapping interactions support fractionalization and generically lead to nontrivial topological order. All of these topological states support fractional gapped (gapless) bulk (respectively, boundary) quasiparticle excitations.