**Chiral Dirac-like fermions**

The Dirac fields obey the famous Dirac equation, \((-i{\alpha }^{i}{\partial }_{i}+m\beta )\psi \left(x\right)=i{\partial }_{0}\psi \left(x\right)\), where \({\alpha }^{i}={\tau }_{x}⨂{\sigma }_{i}\) and \(\beta ={\tau }_{z}⨂{\sigma }_{0}\). With the operators furnishing a four-dimensional irreducible representation of the Lorentz group, the Dirac field can be decomposed into two two-component Weyl fields with opposite chirality in the limit of zero mass. In the context of condensed matter physics, it implies that the chirality of a massless Dirac fermion must be zero because the space-time *PT*-symmetry (*P* and *T* denote space inversion and time reversal, respectively) forces the two branches of each doubly degenerate band to have opposite Berry curvatures (Fig. 1a). Hence, the Fermi arc surface states connecting two Dirac points in a DSM are generally not topologically protected, unlike the Fermi arc connecting chiral Weyl fermions24. On the other hand, chiral fermions with charge-2 chirality have been predicted and measured in materials such as CoSi25–29, of which the band structure manifests four-fold degeneracy node protected by nonsymmorphic symmetry and nondegenerate bands off the high-symmetry point (Fig. 1a).

Recently, it is predicted that two Weyl fields with the same chirality could be connected together to form a “Dirac-like” fermion that fulfills the massless Dirac equation \(i{\alpha }^{i}{\partial }_{i}\psi \left(x\right)=\pm i{\partial }_{0}\psi \left(x\right)\), where \(\psi \left(x\right)\) denotes a four-component free field operator and \({\alpha }^{i}={\tau }_{i}⨂{\sigma }_{0}\) (\({\tau }\) and \({\sigma }\) are two sets of Pauli matrices)14. Therefore, it manifests four-fold degenerate nodes formed by two doubly degenerate bands, while carrying Chern numbers \(C=\pm 2\) (Fig. 1a). Interestingly, the symmetry that connects the two Weyl fields is a counterpart of isospin SU(2) symmetry that relates a proton and a neutron in high-energy physics (see Supplementary S1). In solid state physics, such continuous symmetry does not exist in the framework of conventional (magnetic) crystallographic groups. Instead, the generators of such hidden SU(2) symmetry belong to spin group, which involves partially decoupled spatial and spin operations17, providing a symmetry description of magnetic materials with local moments in the non-relativistic limit. Despite several predicted material candidates, such chiral Dirac-like fermions is either not experimentally observed in quantum materials, or associated with any emergent phenomena. In this work, we provide strong experimental evidence of the existence of such exotic fermions in a chiral antiferromagnet CoNb3S6 through neutron diffraction and angle-resolved photoemission spectroscopy (ARPES), and discuss the nature of the chiral Dirac-like fermions as the source of the unexpected large anomalous Hall effect18.

**Magnetic structure of CoNb** **3** **S** **6**

Our magnetization measurements suggest a phase transition around *T* = 28.3 K, with most of the moments ordered antiferromagnetically in the *ab* plane, and a weak ferromagnetic component along the *c* axis. This observation is consistent with the previous reports18–21. The detailed magnetization data is presented in Supplementary S2, S3 and S4. Figure 1b presents the single-crystal neutron diffraction patterns of CoNb3S6 in the (H, K, 1) and (H, K, 0) scattering plane at 3 K. The magnetic peaks marked by the blue circles, stars and squares can be indexed by three different magnetic wave vectors (0.5, 0, 0), (0, 0.5, 0) and (0.5, -0.5, 0), respectively, indicating three types of magnetic domains rotated \(120^\circ\) from each other. Due to the limited magnetic reflections of the single-crystal diffraction experiment, an additional powder neutron diffraction experiment was performed at 10 K < *T**N* to determine the magnetic structure of CoNb3S6, as shown in Fig. 1c. Consistent with the single-crystal diffraction results, extra weak magnetic reflections are observed and indexed by the same magnetic propagation vector *q**m* = (0.5, 0, 0), (0, 0.5, 0) or (0.5, -0.5 ,0). Representation analysis was applied to analyze the possible magnetic structures30. For the space group *P6**3**22* with Co site at (1/3, 2/3, 1/4) and *q**m* = (0.5, 0, 0), the spin configuration can be described by four different irreducible representations. By Rietveld refinement, we found that Γ4 could give the best fit with *R*p = 2.88 and *R*wp = 3.68 for powder neutron diffraction data. The resulting magnetic moment on the site (1/3, 2/3, 1/4) are antiferromagnetically coupled to the site (2/3, 1/3, 3/4). The detailed analysis of the magnetic structure can be found in Supplementary S3.

As schematically presented in Fig. 1d, the refined magnetic structure of CoNb3S6 shows a collinear magnetic configuration31. By the refinement, we find the Co moments are lying in the *ab* plane and the ordered moment of Co3+ is about 1.24 (14) *µ**B**/*Co. For the intralayer, the local moments on the neighboring Co sites are antiferromagnetically coupled along the *a* axis, but ferromagnetically coupled along the *b* axis.

**Emergence of Fermi-arc surface states**

We perform DFT band structure calculation of bulk CoNb3S6 based on the measured AFM order, as shown in Fig. 1e. Since the bands of interest are dominated by Co 3*d* orbitals with weak SOC, we ignore SOC for the calculations that compare with the ARPES measurement and leave the SOC effects in later discussions. There are two main features in the calculated band structure: Firstly, although *PT* symmetry is absent, the energy bands of any momenta are doubly degenerate. Such degeneracy is unique in magnetic materials without SOC in that it is protected by the so-called spin space group symmetry, which involves independent spin and spatial rotations compared with the conventional magnetic space group15,17. In CoNb3S6, the collinear AFM order guarantees U(1) symmetry along the *x* axis \(\left\{{U}_{x}\left(\theta \right)\right|\left|E\right|0\}\) and a 180\(^\circ\) pure spin rotation along the *z* axis followed by a fractional translation \(\left\{{U}_{z}\left(\pi \right)\right|\left|E\right| {{\tau }}_{({a}+{b})/2}\}\), ensuring doubly degenerate bands throughout the Brillouin zone (see Supplementary S5.1). Secondly, the band crossings are all four-fold degenerate Dirac-like points, which could appear either at arbitrary momenta or high-symmetry lines. There are multiple four-fold Dirac-like points around the calculated Fermi level and some located along \({\Gamma }-\text{X}\) and \({\Gamma }-\text{Y}\) lines (~ 0.4 eV above the Fermi level). Interestingly, unlike the degeneracy protected by *PT*, all the Dirac-like points manifest Chern numbers \(C=\pm 2\) rather than 0, manifesting robust Fermi-arc surface states (see Supplementary S5.2).

We next perform ARPES measurements on the natural cleavage plane (\(ab\) plane) to directly visualize the band structure of CoNb3S6. Although the AFM order enlarges the unit cell leading to a rectangular Brillouin zone (BZ), the ARPES measured spectral intensity exhibits a hexagonal symmetry matching the nonmagnetic 3D BZ (Fig. 2a). This comes from the fact that ARPES spectral intensity averages photoelectrons excited from energy degenerate AFM domains with three different orientations as revealed by the neutron diffraction results. Thus, we use the nonmagnetic 3D BZ to describe the ARPES data measured at \(T=8 K\), *i.e.*, the AFM phase. The general band structure along the high-symmetry line \(\stackrel{-}{{\Gamma }}-\stackrel{-}{K}-\stackrel{-}{M}-\stackrel{-}{{\Gamma }}\) is shown in Fig. 2b. Close to the Fermi level, the ARPES spectra is dominated by a hole-like, highly dispersed band (labeled as\(\alpha\)) centered at \(\stackrel{-}{{\Gamma }}\) and an electron-like, shallow band (labeled as \(\beta\)) centered at \(\stackrel{-}{K}\). To define the precise position of high-symmetry points in the 3D BZ and uncover the \({k}_{z}\) dependence of these bands, we perform photon-energy dependent measurements (\(hv=60 \tilde 165 eV\)), with the momentum cut fixed along \(\stackrel{-}{K}-\stackrel{-}{{\Gamma }}-\stackrel{-}{K}\) direction (Figs. 2c and 2d). As shown in Fig. 2d by the Fermi surface mapping in \({k}_{z}-{k}_{x}\) plane, both \(\alpha\) and \(\beta\) bands show no observable dispersion with \({k}_{z}\), despite some intensity change, consistent with its layered lattice structure. The \({k}_{z}\) periodicity can only be observed if we choose the energy window \(\tilde1 eV\) below the Fermi level and focus on the spectral intensity variation from \(\stackrel{-}{{\Gamma }}\). As shown in Fig. 2c, broad but alternating electron-like and hole-like features can be distinguished as indicated by the superposed white dotted lines. It is noted that the \({k}_{z}\) dispersion shows a \(4\pi /c\) periodicity with lattice constant \(c=11.886 \)Å, because each nonmagnetic unit cell contains two -NbS2-Co1/3-units.

The 2D nature of \(\alpha\) and \(\beta\) bands are further elaborated by examining their dispersion at different \({k}_{z}\) values. As shown in Fig. 2e, we plot \(\stackrel{-}{K}-\stackrel{-}{{\Gamma }}-\stackrel{-}{K}\) cuts from five randomly selected \({k}_{z}\) values, all of which show almost the same dispersion for both \(\alpha\) and \(\beta\) bands. In particular, the Fermi momentum (\({k}_{F}\)) of \(\beta\) band is indicated by the white dashed lines, and it remains constant with \({k}_{z}\), strongly demonstrating its 2D nature. Previously, this electron pocket was attributed to the bulk electronic structure and are dominated by Co atoms22,23, evidenced by the \({k}_{z}\) dispersion observed by ARPES using soft X-ray photons21. Here we use UV photons with much higher energy and momentum resolution to clearly prove its \({k}_{z}\) independence and will discuss its surface origin in the following.

We then compare the ARPES spectra to the projection of DFT calculated bulk and surface bands to fully demonstrate the surface origin of \(\beta\) band and its association with the predicted chiral Dirac-like fermions. Since the structure of CoNb3S6 is indeed stacking NbS2 layers with a Co-layer intercalation, the calculated surface states of NbS2 termination are adopted for comparison. We find that almost all the ARPES measured low-energy band features (including \(\alpha\) and \(\beta\) bands, Fig. 3a, 3d) can be reproduced by DFT projected surface (Fig. 3b, 3e) and bulk calculations (Fig. 3c, 3f). Figure 3a shows the ARPES spectra along \(\stackrel{-}{K}-\stackrel{-}{{\Gamma }}-\stackrel{-}{K}\). The \(\beta\) band centered at \(\stackrel{-}{K}\) comes from an electron pocket with its band bottom slightly below the Fermi level. Such feature can be well reproduced by the DFT-calculated surface states as shown in Fig. 3b. While the previous works suggested that such electron pocket is predominantly from the bulk21, its \({k}_{z}\) independence (Fig. 2d, 2e) and the agreement between ARPES measurement and DFT calculations (Fig. 3a, 3b) synergistically support a surface origin. The ARPES spectra along \(\stackrel{-}{M}-\stackrel{-}{{\Gamma }}-\stackrel{-}{M}\) also reveals weak spectral weight centered at \(\stackrel{-}{M}\) (Fig. 3d). We attribute this feature to the tail of the \(\beta\) surface band which locates slightly above the Fermi level at \(\stackrel{-}{M}\) as shown in Fig. 3e. Such a tail is also visible for the \(\beta\) band at \(\stackrel{-}{K}\), likely from the incoherent electron scattering off other entities such as disorders, bosons and so on32. In Supplementary S5.2, orbital projection analysis shows that \(\beta\) band is dominated by intercalated Co-3*d* atoms. Further calculations show that the \(\beta\) surface pocket are indeed the Fermi-arc surface states originated from the predicted chiral Dirac-like points located 0.17 eV above the Fermi level (see Supplementary S5.2).

While along \(\stackrel{-}{M}-\stackrel{-}{{\Gamma }}-\stackrel{-}{M}\) the ARPES and DFT surface spectra show agreement, there is a slight mismatch of the band edge (minimum) of the \(\beta\) pocket at \(\stackrel{-}{K}\) obtained from ARPES and DFT. Such discrepancy is because our DFT calculations is based on a single-domain collinear AFM order, with the rectangular magnetic BZ (MBZ) shown in Fig. 3g. As a result, the band edge appears at the \(\stackrel{-}{Y}\) point, the boundary of the MBZ. On the other hand, the APRES spectra inevitably averages multiple magnetic domains with the same ground-state energy, thus restoring the hexagonal symmetry for the surface BZ (SBZ) and band edges appearing at the \(\stackrel{-}{K}\) valley (\(\stackrel{-}{{\Gamma }K}=\frac{4}{3}\stackrel{-}{{\Gamma }Y}\)). Figure 3h shows the measured Fermi surface indicating an identical shape and size of BZ to that of a nonmagnetic unit cell. However, the measured Fermi surface as well as the dispersion can hardly be reproduced by the nonmagnetic calculation of CoNb3S6 (see Supplementary S5.3), while the AFM order gives rise to a rectangular MBZ with lower symmetry.

To solve the dilemma, we consider three equivalent q-vectors related by C3 rotational symmetry, rendering three energetically degenerate magnetic domains with three rectangular MBZs rotating 120 degrees with respect to each other. Therefore, an effective hexagonal SBZ is formed, with the size identical to the nonmagnetic one (Fig. 3g). The left panel of Fig. 3i shows the calculated surface state Fermi surfaces for each single AFM domain and illustrates the formation of the hexagonal Fermi surface by superposing the Fermi surfaces of these three equivalent domains. Figures 3h and 3i compare the Fermi surfaces from ARPES and DFT. ARPES mapping reveals six triangular pockets centered at \(\stackrel{-}{K}\) formed by the \(\beta\) pocket (indicated by the black arrow in Fig. 3h). The closed shape of such pocket is in line with the fact that the bottom of \(\beta\) pocket resides below the Fermi level at \(\stackrel{-}{K}\) and above it at \(\stackrel{-}{M}\). According to our DFT calculation, each triangular pocket is formed by three peanut-like surface pockets from three equivalent AFM domains. The broadness of the measured surface bands, as evidenced by the momentum distribution curve analysis in Supplementary S6, may smear out the fine structure of the calculated surface states, resulting into broad \(\beta\) band features centered at \(\stackrel{-}{K}\). The general agreement between ARPES and DFT results throughout this work validates the above arguments and the existence of Fermi arc surface states associated to the chiral Dirac-like fermions in CoNb3S6.

**Discussion of the mysterious large anomalous Hall effect**

As a collinear antiferromagnet, CoNb3S6 exhibits unexpected large AHC below Neel temperature, whose mechanism remains elusive. Previous scenarios ignored the local moments on Co by assuming a nonmagnetic configuration, leading to Weyl points protected by Kramers degeneracy21. Here we report AHE observations by transport measurements and provide systematic calculations based on the scenario of chiral Dirac-like fermions. Figure 4a presents the field evolution of the Hall resistivity measured from 22 K to 29 K with I // *a* and B // *c*. For *T* = 29 K > *T*N, linear dependence of the Hall resistivity as a function of magnetic field (brown line in Fig. 4a) was observed. The positive slope of the Hall resistivity suggests that holes are the majority charge carriers in CoNb3S6. If a single-band model was assumed, the carrier concentration can be estimated by *n* = 1/|*eR*0| = 1.52 × 1021 cm− 3 at *T* = 29 K, where *R*0 is the ordinary Hall coefficient. More detailed information about temperature evolution of carrier concentration is presented in Supplementary S9. As *T* is decreased below *T*N, a pronounced hysteresis builds up. With further lowering the temperature, the hysteresis loop becomes more significant, with the coercive field increasing rapidly. When the temperature is below 23 K, the coercive field becomes larger than 14 T. By subtracting the linear ordinary Hall background and using \({\sigma }_{xy}^{A}={\rho }_{xy}^{A}/({\left({\rho }_{xy}^{A}\right)}^{2}+{\left({\rho }_{xx}\right)}^{2})\), a large anomalous Hall conductivity \({\sigma }_{xy}^{A}\tilde 92 {\left({\Omega }.\text{c}\text{m}\right)}^{-1}\)was obtained at 26 K (Fig. 4b). To examine the ferromagnetic contribution to the anomalous Hall conductivity, the field dependent ferromagnetic component (-*△M*) along the *c* axis is plotted as well (orange empty circles in Fig. 4b). The measured *△M* is ~ 0.001 \({\mu }_{B}\)/Co. In addition, we observed a scaling between the AHE and ferromagnetic canting *△M* in CoNb3S6 as shown in Fig. 4b. Such a strong scaling between the AHE and ferromagnetic canting *△M* could be explained by the large hidden Berry curvature due to the chiral Dirac-like fermions.

Due to the symmetry operation of time-reversal combined with nonsymmorphic translation \(\left\{T\right|\left|E\right|{{\tau }}_{({a}+{b})/2}\}\) (see Supplementary S5.1, Table S4), bulk CoNb3S6 cannot exhibit finite anomalous Hall conductivity. However, rather than being intrinsically absent, the Berry curvature originated from the nontrivial bands are large yet compensated by the global high symmetry33,34. Therefore, the small ferromagnetic (FM) tilting along the *z*-axis and finite SOC play a role of symmetry breaking that reveals the large Berry curvature effect hidden in the otherwise doubly degenerate bands, thus leading to finite anomalous Hall effect. Here we consider bulk and 5-layer thin film CoNb3S6 with the experimental AFM order. Our calculations show that, although under small magnetic tilting, bulk states exhibit AHC smaller than \(1 {\left(cm\bullet {\Omega }\right)}^{-1}\), large AHC approaching 185 \({\left(cm\bullet {\Omega }\right)}^{-1}\) emerges for the thin film, when the chemical potential is 0.01–0.03 eV below the theoretical Fermi level (Fig. 4d). The remarkable difference between bulk and thin film could be attributed to the fact that all the rotation symmetries in thin film CoNb3S6 are broken even without small magnetic tilting (see Supplementary S8, Table S5), as is shown in the crystal structure of CoNb3S6 (Fig. 4c).

There are quite a few chiral Dirac-like fermions near the Fermi level, when considering SOC, some of them split to a twin pair of conventional Weyl points with identical chirality (Fig. 4e) and others are gapped. To reveal the relationship of the chiral Dirac-like fermions and the AHC more clearly, we show another energy window (0.4 ~ 0.6 eV) where there is only one pair of chiral Dirac-like points at 0.48 eV. With SOC and FM tilting, the pair of chiral fermions are gapped, leading to sharp peaks of AHC (~ 200 \({\left(cm\bullet {\Omega }\right)}^{-1}\)) around the corresponding chemical potential (Fig. 4f). Therefore, the large AHE in collinear AFM CoNb3S6 results from the Berry curvature of the chiral Dirac-like fermions as well as symmetry breaking by SOC, ferromagnetic canting. Furthermore, our results are also consistent with the fact that AHC measured in thin films is much larger than that measured in thick slabs19 without initializing a noncollinear magnetic structure unproved experimentally.

Except for above mechanism, due to the complexity of magnetic configuration and surface topography in CoNb3S6, other mechanisms cannot be fully ruled out. One is the surface enhanced AHC. While the measurements of magnetic configuration reflect the reality of the bulk, the surface magnetic structure might be affected by imperfection and reconstruction, leading to distinct geometric structure or magnetic configuration. A typical example is the gapless surface states of MnBi2Te435. In addition, owing to the absence of C2x symmetry for thin film CoNb3S6, the *z* component of the magnetic moment is allowed to exist without breaking any symmetries. Such FM canting might also cause the large AHC in CoNb3S6. Moreover, due to the in-plane C3 symmetry, in addition to the scenario of three equivalent magnetic domains, the compound could also have a triple-*q* magnetic order36 (more detail could be found in Supplementary S3), which may lead to a chiral spin structure and thus the nonzero AHC. In summary, our work evidences the existence of chiral Dirac-like fermion in a topological semimetal CoNb3S6, paving an avenue for exploring unexpected emergent phenomena in collinear antiferromagnets.