Chiral crossings with local and global symmetry constraints

The number of surface states as well as the electric response of Weyl semimetals is determined to a large extent by the Chern number of the Weyl points, a property also known as their chirality. It is known [1,2] that rotation eigenvalues affect the value of the chirality, but so far there is no comprehensive theory that connects rotation eigenvalues and topological charges. Here, we show that the previous works are applications of a local constraint, which holds for arbitrary combinations of symmetries as well as for chiral crossings comprising more than two bands. Using this constraint we explain the chiralities of quadruple Weyl points. Furthermore, with a global constraint, stemming from the periodicity of the Brillouin zone, we can identify enforced topological nodal planes as well as Weyl points at generic positions.

[1] Chen Fang, et al., Phys. Rev. Lett. 108, 266802 (2012)

[2] Stepan S. Tsirkin, et al., Phys. Rev. B 96, 045102 (2017)