Non-Abelian Statistics in Momentum Space

QuanSheng Wu, Alexey A. Soluyanov, Tomáš Bzdušek

Published 2018-08-21Version 1

We introduce non-Abelian statistics in momentum space of crystalline solids with weak spin-orbit coupling. We show that nodal lines and nodal chains of $\mathcal{P}\mathcal{T}$-symmetric metals host non-Abelian quaternion charges, similar to vortices of biaxial nematics. Starting from two-band considerations, we develop a complete many-band description of nodes in the presence of $\mathcal{P}\mathcal{T}$ and mirror symmetries, which explains possible topological phase transitions between different arrangements of nodal lines. Our arguments are illustrated with ${\boldsymbol{k}}\cdot{\boldsymbol{p}}$ models as well as with real materials. We find and explain novel topological invariants that correspond to the quaternion charges. We show that these invariants characterize 1D topological insulator phases not described before. This work introduces a fundamentally novel approach to nodal topological excitations in metals.