## 3 Anomalous Hall Effect in Ferromagnetic Topological Materials
The Weyl points act as a sink and a source of the Berry curvature, just like monopoles in reciprocal space, therefore the \(k_{x}-k_{y}\) planes host a non-zero quantized anomalous Hall conductance when \(k_{z}\) is lying between the pair of Weyl points, and the contribution is zero for the planes with \(k_{z}>|\sqrt{M_{0}/M_{1}}|\)[15; 16]. As a result, the AHC for the magnetic WSMs with one pair of Weyl point is \(\sigma^{z}=\frac{\epsilon^{2}}{h}\frac{k_{w}}{\pi}\) and its magnitude is decided by the separation of the Weyl points in \(k\) space. Because the time reversal symmetry is broken due to the magnetization along \(z\), only the \(z\)-component of the Berry curvature shows a non-zero value after the integral over the whole \(k\) space (see Figure 2d). In contrast, the other two components still follow time reversal symmetry, as shown in Figure 2e,f, and therefore they do not contribute to the AHE. Figure 1: Schematic of a quantum anomalous Hall insulator (QAHI) with a nodal line band structure. (**a**) The band inversion forms a nodal ring in the situation without including SOC. The Berry curvature is zero in the whole \(k\) space. (**b**) The nodal line is broken by SOC with opening a band gap, and non-zero Berry curvature forms a hot ring located in the band gap. The color bar for the Berry curvature is in arbitrary units. (**c**) A quantized anomalous Hall conductance (\(e^{2}/h\)) in the band gap. Figure 2: AHE in magnetic WSMs. (**a**) Energy dispersion of a magnetic WSM with one pair of Weyl points lying on the \(z\)-axis and at zero energy. (**b**) Energy dependent AHC for the effective model of a magnetic WSM. (**c**) Evolution of the anomalous Hall conductance for each \(k_{x}-k_{y}\) plane with varying \(k_{z}\). (**d–f**) Berry curvature distribution in 3D BZ for the three components of \(\Omega^{z}\), \(\Omega^{y}\), and \(\Omega^{x}\). The hot spots are mainly focused around the Weyl points. The color bar is in arbitrary units. Because the nodal lines and Weyl points can generate strong local Berry curvature, magnetic nodal line semimetals and WSMs are expected to have a strong intrinsic AHE. Recently, a large AHE was observed in the Heusler compound Co\({}_{2}\)MnGa [56; 57] and the quasi-two-dimensional compounds Co\({}_{3}\)Sn\({}_{2}\)S\({}_{2}\)[58; 59] and Fe\({}_{3}\)GeTe\({}_{2}\)[60], where the mirror symmetry protected nodal line band structure is the most important feature for this phenomenon. Particularly, owing to the topological band structures and low charge carrier density, Co\({}_{3}\)Sn\({}_{2}\)S\({}_{2}\) and Co\({}_{2}\)MnGa are so far the only two examples of all known materials hosting both giant AHC and anomalous Hall angle (AHA) [57; 58; 59]. Taking Co\({}_{3}\)Sn\({}_{2}\)S\({}_{2}\) as an example, the Co atoms form Kagome lattices stacked along the \(c\) direction (see Figure 3a). Here, the crucial symmetries are the three mirror planes parallel to the \(c\) direction, which results in three pairs of nodal lines connected by a \(C_{3}^{2}\) rotation symmetry, as presented in Figure 3c,e. Because the magnetization is aligned along the \(z\) direction, the mirror symmetries are broken by SOC, and the nodal lines are not longer protected. Therefore, a band gap opens. Meanwhile, one pair of Weyl points with opposite chirality remains along each of the former nodal lines (see Figure 3d). The Berry curvature distribution in \(k\) space is dominated by the SOC-gapped nodal line (see Figure 3e), leading to an AHC of \(\sim\)1100 S/cm. Together with the low charge carrier density, the AHA can reach up to \(\sim\)20%, which is a new record in 3D materials [58; 59].