Accidental Degeneracy in k-space, Geometrical Phase, and the Perturbation of $\pi$ by Spin-orbit Interactions

Since closed lines of {\it accidental} electronic degeneracies were demonstrated to be possible, even frequent, by Herring in 1937, no further developments arose for eight decades. The earliest report of such a nodal loop in a real material – aluminum – is recounted and elaborated on. Nodal loop semimetals have become a focus of recent activity, with emphasis on other issues. Band degeneracies are, after all, the origin of topological phases in crystalline materials. Spin-orbit interaction lifts accidental band degeneracies, with the resulting spectrum being provided here. The geometric phase $\gamma (C)=\pm \pi$ for circuits $C$ surrounding a line of such degeneracy cannot survive completely unchanged. The change depends on how the spin is fixed during adiabatic evolution. For spin fixed along the internal spin-orbit field, $\gamma (C)$ decreases to zero as the circuit collapses around the line of lifted degeneracy. For spin fixed along a perpendicular axis, the conical intersection persists and $\gamma (C)=\pm \pi$ is unchanged.