## 4 Anomalous Hall Effect in Antiferromagnetic Topological Semimetals  
From the symmetry point of view, the AHE in ferromagnets is easy to understand due to the net magnetization, and the important insight is that the special topological band structure can enhance the signal dramatically. However, for a long time, it was believed that the AHE cannot exist in antiferromagnets (AFMs), because the measured Hall resistivity normally follows \(\rho_{Hall}=R_{0}\mu_{0}H+(\alpha\rho_{xx}^{2}+\beta\rho_{xx})\), where the fist term is the ordinary Hall from the Lorentz force and the second term is the anomalous component from extrinsic scattering and intrinsic contributions. Figure 3: AHE in the magnetic WSM Co\({}_{3}\)Sn\({}_{2}\)S\({}_{2}\). (**a**) Crystal and magnetic structure for Co\({}_{3}\)Sn\({}_{2}\)S\({}_{2}\). The three mirror planes are represented by the green lines. (**b**) Energy dependent AHC. (**c**,**d**) Energy dispersion along high symmetry lines for the cases without and with SOC, respectively. The local energy dispersion for the nodal line and the Weyl points is shown in the inset. The green line represents the nodal line. The blue and red points represent Weyl points. (**e**) Berry curvature distribution in the 3D BZ. The color bar is in arbitrary units. Because the intrinsic contribution can be understood from the integral of the Berry curvature in \(k\) space, and the Berry curvature is odd with respect to time reversal symmetry, the AHE can only exist in systems with broken time reversal symmetry. In most collinear AFMs, although the time reversal symmetry \(\hat{T}\) is broken, it is possible to find a combined symmetry of \(\hat{T}\) and a space group operation \(\hat{O}\), which is preserved in the system and still reverses the sign of Berry curvature. Therefore, the AHE is forced to zero in these systems. However, such joint symmetry \(\hat{T}\hat{O}\) does not necessarily exist in all AFM systems. Two counter-examples are non-collinear AFMs and compensated ferrimagnets. The existence of an AHE in AFMs was first proposed by [PERSON] and [PERSON] in 2001 by predicting a non-zero AHE in non-coplanar AFM face-centered-cubic lattices via Berry curvature analysis [61]. Further, it was predicted in 2014 via Berry curvature calculations in cubic Mn\({}_{3}\)Ir [62] and hexagonal Mn\({}_{3}\)\(X\) (\(X\) = Ga, Sn or Ge) [63] that the intrinsic AHE can even exist in coplanar non-collinear AFMs. The latter was experimentally observed in Hall measurements soon after the theoretical prediction [64; 65]. Recently, it was found that the AHE in Mn\({}_{3}\)Ge and Mn\({}_{3}\)Sn is strongly dependent on the symmetry of the magnetic structure and the topological band structure [66]. As presented in Figure 4a, the triangular lattices formed by Mn are stacked along the \(c\) direction, and the magnetic structure follows a glide mirror symmetry \(\left\{M_{y}|(0,0,1/2)\right\}\), see Figure 4b. Because the Berry curvature is invariant with respect to glide operations, the shape of the AHC tensor is only decided by the mirror \(M_{y}\). \(M_{y}\) is similar to a time reversal operation for the \(x\) and \(z\) components of the spin and will change the sign of \(\Omega^{x}\) and \(\Omega^{z}\) while keeping \(\Omega^{y}\) invariant. As a result, the AHC vector takes the form \(\overrightarrow{\sigma}^{\prime}{}^{\prime}{}^{\prime}{}^{\prime}{}^  
Apart from a non-collinear magnetic structure, the combined symmetry that reverses the sign of the Berry curvature can also be broken in collinear magnetic structure with zero net moment. As presented in Figure 4d, the sign of the magnetic moments is changed by the combined time reversal and translation symmetry. This translation symmetry can be broken by an atomic replacement while keeping local moments invariant (see Figure 4e). In this case, the system can obtain a non-zero net Berry curvature and therefore a finite intrinsic AHC. This kind of magnetic structure can exist in compensated ferrimagnets.