Pseudospin Vortex Ring with a Nodal Line in Three Dimensions

We present a model of a topological semimetal in three dimensions (3D) whose energy spectrum exhibits a nodal line acting as a vortex ring; this in turn is linked by a pseudospin structure akin to that of a smoke ring. Contrary to a Weyl point node spectrum, the vortex ring gives rise to skyrmionic pseudospin patterns in cuts on both sides of the nodal ring plane; this pattern covers the full Brillouin zone, thus leading to a new, `maximal', anomalous Hall effect in a 3D semimetal. Tuning a model parameter shrinks the vortex ring until it vanishes, giving way to a pair of Weyl nodes of opposite chirality. This establishes a connection between two distinct momentum-space topologies - that of a vortex ring (a circle of singularity) and a monopole-anti-monopole pair (two point singularities). We present the model both as a low-energy continuum and a two-band tight-binding lattice model. Its simplicity permits an analytical computation of its Landau level spectrum.