Fig. 4: Schematic transition between nodal rings and Weyl points.
a The nodal ring (solid ring) is protected by the in-plane mirror symmetry where the spin (arrow) points out of plane, in the upper panel. If the spin rotates to break the mirror symmetry, the nodal ring gets gapped out (dashed ring), giving rise to a pair of Weyl points (blue spheres). The induced AHE is proportional to the separation of the Weyl points, that is, the diameter of the nodal ring (kd) in a form \(\frac{{e}^{2}}{h}\frac{{k}_{\mathrm{d}}}{2\pi a}\), where a is the lattice parameter. b The nodal ring is as large as the Brillouin zone size, as a critical point. The resultant Weyl point is pushed to the zone boundary and meets another Weyl point with opposite chirality from the second zone. The induced AHE is quantized to \(\frac{{e}^{2}}{h}\frac{1}{a}\). c The nodal ring is larger than the Brillouin zone size, as an open nodal ring. Then, the gapped ring does not induce Weyl points in the open direction, generating a 3D quantized Hall conductivity, \(\frac{{e}^{2}}{h}\frac{1}{a}\).
