Fig. 3: The anomalous Hall effect and topological band structure.

a The anomalous Hall conductivity \({\sigma }_{{yx}}^{A}\) as rotating the magnetization from the z-axis to y-axis. The experimental and theoretical results are represented by red and black circles, respectively. The solid curve is a cosine profile to guide eyes. b Nodal rings and the first Brillouin zone of Co₂MnAl. Without SOC, there are nodal rings in these mirror planes, k_x,y,z = 0 or π. For example, there are four nodal rings that are labeled as #1–4 in c, centered at the Z point of the FCC Brillouin zone. When magnetization align along the [001] direction by SOC, all nodal rings are gapped out because of the mirror symmetry breaking, except those in the k_z = 0 or π plane. The preserved nodal rings and related band structure are shown in c. The band dispersion, in which the nodal ring crossing points are indicated by blue dots, is along the black dotted line in the upper panel. d–f The Berry-curvature (Ω_yx) distribution in the k_x = 0 plane centered at Γ for different magnetization angle (θ). Nodal rings (black) are gapped out in this plane. Near the charge neutral point, the Berry curvature is mainly contributed by the gapped nodal ring #3. When θ = 45°, nodal rings #1 and #4 evolve into Weyl points (filled black circles in e), while nodal rings #2 and #3 are fully gapped. g The nodal ring #3 at four corners can be treated as the reconstruction of a large ring, which is illustrated by a green dashed circle, centered at Γ.