Beyond the Tenfold Way: Non-Abelian Topology in Noninteracting Metals

Abstract

We introduce non-Abelian topological charges for nodal-line band degeneracies in momentum space of PT-symmetric crystalline metals with weak spin-orbit coupling. We show that these are quaternion charges, similar to those describing vortices in biaxial nematics. Starting from two-band considerations, we develop the complete many-band description of nodes in the presence of PT and mirror symmetries. This theory allows us to investigate the topological stability of nodal chains in metals. The non-Abelian charges put strict constraints on the possible nodal line compositions and the possibility to annihilate them. Our arguments are illustrated with k⋅p models as well as with real materials. Our analysis goes beyond the "tenfold way" approach to band topology, and implies the existence of 1D topological phases not present in the "tenfold way" classifications.