## 3 Anomalous Hall Effect in Ferromagnetic Topological Materials  
Considering the observable \(\hat{A}\) as electrical current density \(\hat{A}=-e\hat{v}/V\), the inter-band part of linear response tensor for each separated \(k-\)point becomes the so-called Berry curvature [42; 43]  
\[\Omega^{k}(n,k)=\sum_{m\
eq n}\frac{\langle n,k|\hat{v}_{i}|m,k\rangle\langle m,k|\hat{v}_{j}|n,k\rangle}{(E_{n,k}-E_{m,k})^{2}} \tag{7}\]and the anomalous Hall conductivity (AHC) is dependent on the integral of the Berry curvature over the whole Brillouin zone (BZ). Because the Berry curvature is odd with respect to time reversal symmetry, the anomalous Hall effect (AHE) can only exist in magnetic systems in the linear response approximation. The AHE was first observed in 1879 by [PERSON] in two-dimensional ferromagnetic systems [44]. According to the current understanding, there are mainly two contributions for the AHE: one is the extrinsic contribution from impurity scattering and the other is the intrinsic contribution from Berry phase effects of the electronic band structure of the ideal single crystal [42; 43; 45; 46; 47]. Because both symmetry and topology are tied to the ideal crystal, it is not possible to discuss the the extrinsic contribution from the topological point of view. In the following, we therefore focus on the intrinsic part. In topological materials, owing to the special band structure, the local Berry curvature is strongly enhanced, and the AHE is dominated by the intrinsic contribution. The extreme example is the quantum anomalous Hall effect (QAHE), where only intrinsic contributions exist, with a quantized transverse anomalous Hall conductance in quanta of \(e^{2}/h\) and zero longitudinal resistance [48]. The topological charge in the QAHE is indexed by a topological invariant, the Chern number, which is given by the integral of the Berry curvature over the whole 2D BZ.In contrast to the quantum Hall effect, which is decided by the external magnetic field [49], the QAHE originates from the spin-orbit coupling (SOC) and intrinsic magnetism. The QAHE was theoretically proposed by [PERSON] in 1988 by a model defined on a honeycomb lattice [48]. Owing to the development of topological band theory and thin film growth techniques, the first QAHE was experimentally observed in chromium-doped (Bi,Sb)\({}_{2}\)Te\({}_{3}\)[50; 51]. An effective model for the QAHE can be described by a \(2\times 2\) nodal line band structure [52]  
\[H_{QAHE}=A(k_{x}\sigma_{x}+k_{y}\sigma_{y})+M(k)\sigma_{z}, \tag{8}\]  
where \(\sigma_{i}\) are the Pauli matrices and \(M(k)=M_{0}+M_{1}(k_{x}^{2}+k_{y}^{2})\). Because this model includes a mirror symmetry with respect to \(x-y\) plane, the band structure forms a nodal loop in the absence of SOC, as presented in Figure 1a. This nodal line is broken by SOC and a band gap opens, which generates non-zero Berry curvature in the band gap and forms a hot loop in 2D \(k\) space (see Figure 1b). The integral of the Berry curvature in the whole 2D BZ leads to a quantized anomalous Hall conductance in the band gap, which is just corresponding to the chiral edge state, as presented in Figure 1c. A stack of 2D QAHE systems was proposed to be an effective way to obtain magnetic Weyl semimetals (WSMs) with the QAHE in its 3D version [53; 54; 55]. The simplest model can be obtained by including the coupling effect along \(z\) via adding a \(k_{z}\) term \(M_{1}k_{z}^{2}\sigma_{z}\) into Equation (8) [55], where one pair of Weyl points is located on the \(k_{z}\)-axis at \((0,0,\pm k_{w})\) with \(k_{w}=\sqrt{M_{0}/M_{1}}\), as presented in Figure 2a. The peak value of the anomalous Hall conductivity (AHC) lies at the energy of the pair of Weyl points (Figure 2b). The AHC in magnetic Weyl semimetals can be understood from the QAHE.