Topological aspects of nodal-loop semimetals and the nature of spin-orbit coupling in Lu₂V₂O₇

This thesis is presented in two parts: (I) understanding the topological aspects of nodal-loop semimetals in the presence of interactions and (II) exploring the nature of spin-orbit coupling in the pyrochlore oxide Lu₂V₂O₇. Each of these parts correspond to distinct research projects, sharing an underlying theme of topology and spin-orbit coupling. (I) Topological nodal-loop semimetals are characterized by a symmetry protected 1D line of band touching points, dispersionless surface states, and an electromagnetic response in the form of an induced magnetization and polarization. While this characterization holds true for free fermions, there has recently been increased interest in how topological order manifests in the presence of interactions, particularly in the gapless topological semimetals. This thesis explores the effect of interactions in nodal-loop semimetals, in the framework of a mean-field BCS theory. It is found that the surface states are preserved for p-wave pairing when the gap function preserves the symmetries that protect the nodal-loop. Furthermore, it is shown that this state hosts a single Dirac mode in the core of any vortex line of odd vorticity. This result highlights that the topological order of a nodal-loop can remain stable in the presence of interactions. (II) Pyrochlore oxides geometrically support a magnetic interaction of spin-orbit origin known as the Dzyaloshinskii-Moriya interaction, due to the lack of inversion symmetry between neighbouring magnetic ions. In magnetically ordered quantum systems, such an interaction has been shown to give rise to a thermal Hall effect of magnons, the quasiparticles arising from spin fluctuations. Such an effect was subsequently observed in thermal transport measurements of the pyrochlore oxide Lu₂V₂O₇, however, these measurements seem to overestimate the magnitude of the DM interaction. This draws into question the nature of spin-orbit coupling in this material. An overview of the low-energy properties of Lu₂V₂O₇ is presented, along with a re-investigation of the thermal transport data. This leads to an apparent discrepancy in the value of the DM interaction, motivating future investigations of this material