Non-Abelian reciprocal braiding of Weyl nodes and fragile topology

Weyl nodes trapped within C₂T symmetric planes (i.e. -crystalline rotation combined with time reversal) acquire a non-Abelian topological charge on top of their chirality. E.g. three-level systems realize the quaternion group [1]. The non-Abelian nature of Weyl nodes can alternatively be captured by the Euler class [2,3], which itself can be efficiently computed through the Wilson loop as the winding number of a Pfaffian [3]. These tools allow us (i) to design minimal experimental setups, and (ii) to search for material candidates. Furthermore, we show that additional crystalline symmetries can lead to the obstruction of the Euler class which naturally gives rise to nontrivial fragile topology.

[1] Q.-S. Wu, A. A. Soluyanov, and T. Bzdušek, Science 365, 1273 (2019)
[2] J. Ahn, S. Park, and B-J Yang, Phys. Rev. X 9, 02013 (2019)
[3] A. Bouhon, R.-J. Slager, and T. Bzdušek, arXiv:1907.10611 (2019)