Non-Abelian band topology in noninteracting metals

Electron energy bands of crystalline solids generically exhibit degeneracies called band-structure nodes. Here, we introduce non-Abelian topological charges that characterize line nodes inside the momentum space of crystalline metals with space-time inversion ($\mathcal{PT}$) symmetry and with weak spinorbit coupling. We show that these are quaternion charges, similar to those describing disclinations in biaxial nematics. Starting from two-band considerations, we develop the complete many-band description of nodes in the presence of $\mathcal{PT}$ and mirror symmetries, which allows us to investigate the topological stability of nodal chains in metals. The non-Abelian charges put strict constraints on the possible nodal-line configurations. Our analysis goes beyond the standard approach to band topology and implies the existence of one-dimensional topological phases not present in existing classifications.