Magnetic Weyl semimetals Weyl semimetals (WSMs)—materials that host exotic quasiparticles called Weyl fermions—must break either spatial inversion or time-reversal symmetry. A number of WSMs that break inversion symmetry have been identified, but showing unambiguously that a material is a time-reversal-breaking WSM is tricky. Three groups now provide spectroscopic evidence for this latter state in magnetic materials (see the Perspective by da Silva Neto). Belopolski et al. probed the material Co 2 MnGa using angle-resolved photoemission spectroscopy, revealing exotic drumhead surface states. Using the same technique, Liu et al. studied the material Co 3 Sn 2 S 2 , which was complemented by the scanning tunneling spectroscopy measurements of Morali et al. These magnetic WSM states provide an ideal setting for exotic transport effects. Science, this issue p. 1278, p. 1282, p. 1286; see also p. 1248

Abstract Topological matter is known to exhibit unconventional surface states and anomalous transport owing to unusual bulk electronic topology. In this study, we use photoemission spectroscopy and quantum transport to elucidate the topology of the room temperature magnet Co 2 MnGa. We observe sharp bulk Weyl fermion line dispersions indicative of nontrivial topological invariants present in the magnetic phase. On the surface of the magnet, we observe electronic wave functions that take the form of drumheads, enabling us to directly visualize the crucial components of the bulk-boundary topological correspondence. By considering the Berry curvature field associated with the observed topological Weyl fermion lines, we quantitatively account for the giant anomalous Hall response observed in this magnet. Our experimental results suggest a rich interplay of strongly interacting electrons and topology in quantum matter.

Topological phases of matter can host analogs of particles relevant for high-energy physics and may enable fresh pathways for the application of quantum materials (1–10). To date, most topological phases have been discovered in nonmagnetic materials (6–8), thus limiting their magnetic field tunability, as well as their electronic and magnetic functionality. Identifying and understanding electronic topology in magnetic materials has the potential to lead to the discovery of novel magnetic responses that may be used for future spintronics technologies. Recently, several magnets were found to exhibit a large anomalous Hall response in transport, which has been linked to a large Berry curvature in their electronic structures (11–15). However, it remains largely unclear in experiment whether the Berry curvature originates from a topological band structure, such as Dirac and Weyl points or node lines. In particular, there is no direct visualization of a topological magnetic phase demonstrating a bulk-boundary correspondence with associated anomalous transport. In this study, we use angle-resolved photoemission spectroscopy (ARPES), ab initio calculation, and transport to explore the electronic topological phase of the ferromagnet Co 2 MnGa (10). We observe node lines that we identify as Weyl fermion lines, along with drumhead surface states. We find that the Weyl lines concentrate the Berry curvature, accounting for the giant intrinsic anomalous Hall response in Co 2 MnGa.

Topological phases can be categorized by the dimensionality of the band touching into topological insulators, node point semimetals, and node line semimetals (8, 16–18). Node points are often further subcategorized as Dirac points, Weyl points, and other exotic point touchings (8). Analogously, node lines may include Dirac lines (fourfold degenerate), Weyl lines (twofold degenerate), and possibly other one-dimensional (1D) band crossings (19–21). Node lines may be protected by crystal mirror symmetry, giving rise to drumhead surface states (9, 17, 18, 22–25). Co 2 MnGa takes a cubic face-centered Bravais lattice (Fig. 1A), space group F m 3 ¯ m (no. 225), indicating the presence of several mirror symmetries in the system. Moreover, the material is ferromagnetic with Co and Mn moments (26) and Curie temperature T C = 690 K (Fig. 1B) (27), indicating broken time-reversal symmetry. This suggests that all bands are generically singly degenerate and that mirror symmetry may give rise to twofold-degenerate node lines. In a detailed theoretical analysis, we studied the band structure of Co 2 MnGa by ab initio calculation, neglecting spin-orbit coupling (SOC). We found that the ferromagnetic exchange splitting drives a phase with two majority spin bands near the Fermi level that exhibit twofold degeneracies on the mirror planes (18). These degeneracies, which arise from a crossing of bands with opposite mirror eigenvalues, form three families of Weyl lines (Fig. 1C and fig. S10), which are pinned to each other, forming a nodal chain; some of them further form Hopf-like links with one another (10). The predicted Weyl lines are protected only when the SOC is strictly zero, but numerical results in the presence of SOC suggest that the gap opened is negligible (fig. S9).

Fig. 1 Magnetic node line in Co 2 MnGa. (A) Crystal structure of Co 2 MnGa. (B) Magnetization as a function of temperature of Co 2 MnGa single crystals, in the absence of a magnetic field [zero-field–cooled (ZFC)] and cooled under a constant magnetic field of μ 0 H = 200 Oe oriented along the [001] crystallographic axis [field-cooled (FC)]. We find a Curie temperature T C = 690 K. μ B , the Bohr magneton; f.u., formula unit. (C) Schematic of a generic node line. A node line (red curve) is a band degeneracy along an entire curve in the bulk Brillouin zone. It is associated with a drumhead surface state stretching across the node line (green sheet). In the case of a mirror-symmetry–protected node line, the node line lives in a mirror plane of the Brillouin zone but is allowed to disperse in energy. (D to H) Constant-energy surfaces of Co 2 MnGa measured by ARPES at hν = 50 eV and T = 20 K, presented at a series of binding energies, E B , from the Fermi level, E F , down to E B = 0.08 eV. (I) Schematic of constant-energy cuts (green) of a node line, suggesting a correspondence with the observed ARPES dispersion.

Motivated by these considerations, we investigate Co 2 MnGa single crystals by ARPES. We focus first on the constant-energy surfaces at different binding energies, E B . We readily observe a feature that exhibits an unusual evolution from a < shape (Fig. 1, D and E) to a dot (Fig. 1F) to a > shape (Fig. 1, G and H). This feature suggests that we observe a pair of bands that touch at a series of points in momentum space. As we shift downward in E B , the touching point moves from left to right (black guides to the eye in Fig. 1, D to H), and at certain E B (Fig. 1F) the spectral weight appears to be dominated by the crossing point. This series of momentum-space patterns is characteristic of a node line (Fig. 1I). For the constant-energy surfaces of a node line, as we slide down in E B , the touching point slides from one end of the node line to the other, gradually zipping closed an electron-like pocket (upper band) and unzipping a hole-like pocket (lower band). To better understand this result, we consider E B − k x (k x , in-plane crystal momentum) cuts passing through the node line feature (Fig. 2A). On these cuts, we observe a candidate band crossing near k x = 0 . We further find that this crossing persists in a range of k y and moves downward in energy as we cut further from Γ ¯ (more negative k y ). We can fit the candidate band crossing with a single Lorentzian peak, suggestive of a series of touching points between the upper and lower bands (fig. S13). Taking these fitted touching points, we can in turn fit the dispersion of the candidate node line to linear order, obtaining a slope v = 0.079 ± 0.018 eVÅ . Lastly, we observe that at a given k y , the bands disperse linearly in energy away from the touching points. Taken together, our ARPES results suggest the presence of a node line at the Fermi level in Co 2 MnGa.

Fig. 2 Evidence for a Weyl fermion line. (A) Series of ARPES E B − k x cuts through the candidate node line, corresponding to the feature discussed in Fig. 1. The band crossing points near k x = 0 are fit with a single Lorentzian peak (cyan dots), and the train of dots is then fit with the experimentally observed node line dispersion (blue line). (B) Ab initio E B − k x predicted bulk bands of Co 2 MnGa in the ferromagnetic state, projected on the (001) surface, predicting a Weyl line at k x = 0 (white arrow) (10). The colors indicate the spectral weight of a given bulk state on the surface, obtained using an iterative Green’s function method (see supplementary materials and methods) (28, 33). (C) Same as Fig. 1E, with the (001) surface Brillouin zone marked (green box). (D) Key features of the data, obtained from analysis of the momentum and energy distribution curves of the ARPES spectra. (E) Ab initio constant-energy surface at binding energy E B = 0.08 eV below E F , on the (001) surface with MnGa termination, showing qualitative agreement with the ARPES, as marked by features a to d. (F) Projection of the predicted Weyl lines on the (001) surface, with energy axis collapsed, suggesting that the key features observed in ARPES and DFT arise from the predicted Weyl lines: a (blue), b (red), c (yellow), d (another copy of the yellow Weyl line).

To better understand our experimental results, we compare our spectra with ab initio calculations of Co 2 MnGa in the ferromagnetic state (10). We consider the spectral weight of bulk states on the (001) surface and study an E B − k x cut in the region of interest (Fig. 2B) (28). At k x = 0 , we observe a band crossing (white arrow), which we can trace back in numerics to a node line near the X point of the bulk Brillouin zone (blue node line in fig. S10). According to our earlier theoretical analysis, the node line resides on the M xy (and equivalent) mirror planes (10). This node line is a Weyl line, in the sense that it is a twofold-degenerate band crossing extended along one dimension (19–21). It is predicted to be pinned to a second, distinct Weyl line, forming part of a nodal chain. To compare experiment and theory in greater detail, we plot the calculated dispersion of the Weyl line against the dispersion as extracted from Lorentzian fits of ARPES data (fig. S15). We observe a hole-doping of experiment relative to theory of E B = 0.08 ± 0.01 eV . We speculate that this shift may be caused by a chemical doping of the sample or an approximation in the way that density functional theory (DFT) captures magnetism in this material. The correspondence between the crossing observed in ab initio calculation and ARPES suggests that we have observed a magnetic Weyl line in Co 2 MnGa.

Having considered the blue node line marked in Fig. 2A, we search for other node lines in our data. We compare an ARPES spectrum (Fig. 2, C and D) to an ab initio calculation of the surface spectral weight of bulk states, taking into account the observed effective hole-doping of our sample (Fig. 2E). In addition to the blue Weyl line (labeled in Fig. 2, D and E, as a), we observe a correspondence between three other features in experiment and theory, marked as b, c, and d. To better understand the origin of these features, we consider all of the predicted Weyl lines in Co 2 MnGa (10) and plot their surface projection with the energy axis collapsed (Fig. 2F). We observe a correspondence between b in the ARPES spectrum and the red Weyl line. Similarly, we see that c and d match with predicted yellow Weyl lines. To further test this correspondence, we look again at our ARPES constant-energy cuts and find that d exhibits a transition from a < shape to a > shape, which suggests a node line (figs. S16 and S17). The comparison between ARPES and ab initio calculation suggests that an entire network of magnetic Weyl lines is realized in Co 2 MnGa.

Next we explore the topological surface states. We study the ARPES spectrum along k a , as marked by the green line in Fig. 3F. On this cut we observe three cones (red arrows in Fig. 3A) that are consistent with the yellow Weyl lines. Notably, we also observe a pair of states that appear to connect one cone to the next (Fig. 3, A to C). Moreover, these extra states consistently terminate on the candidate yellow Weyl lines as we vary k b (fig. S20). We further carry out a photon energy dependence study and discover that these extra states do not disperse with photon energy from hν = 34 to 48 eV (h, Planck’s constant; ν, frequency), suggestive of a surface state (Fig. 3G). In ab initio calculation, we observe a similar pattern of yellow Weyl lines pinning a surface state (Fig. 3E) (28). These observations suggest that we have observed a drumhead surface state stretching across Weyl lines in Co 2 MnGa. The pinning of the surface states to the cones further points to a bulk-boundary correspondence between the bulk Weyl lines and the drumhead surface state dispersion.

Fig. 3 Topological drumhead surface states in Co 2 MnGa. (A to C) ARPES E B − k a cuts at different photon energies. We observe three cone-like features [red arrows in (A)]. (D) Key features of the data in (C), obtained from analysis of the momentum and energy distribution curves. Apart from the cone-like features (yellow), there are additional states (green) connecting the cones. (E) The corresponding E B − k a cut from ab initio calculation, crossing three Weyl lines (red arrows) connected by drumhead surface states (see supplementary materials and methods) (28). (F) Same as Fig. 2F, marking the location of the ARPES spectra in (A) to (C) (green line) and defining the k a , b axes. (G) Photon energy dependence of an energy distribution curve passing through the candidate drumhead state [red dashed line in (A)]. The peaks marked by the black vertical line correspond to the drumhead state. We observe no dispersion as a function of photon energy, providing evidence that the candidate drumhead is a surface state.

Now that we have provided spectroscopic evidence for a magnetic bulk-boundary correspondence in Co 2 MnGa, we investigate the relationship between the topological node lines and the anomalous Hall effect (AHE). We study the Hall conductivity σ x y under magnetic field μ 0 H at different temperatures T and extract the anomalous Hall conductivity σ AH ( T ) (Fig. 4A). We obtain a very large AHE value of σ AH = 1530 ohm − 1 cm − 1 at 2 K, consistent with earlier reports (11, 12). To understand the origin of the large AHE, we study the scaling relation between the anomalous Hall resistivity, ρ AH , and the square of the longitudinal resistivity, ρ x x 2 , both considered as a function of temperature. It has been shown that under the appropriate conditions, the scaling relation takes the form ρ AH = ( α ρ x x 0 + β ρ x x 0 2 ) + γ ρ x x 2 where ρ x x 0 is the residual longitudinal resistivity, α is the contribution from skew scattering, β is the side-jump term, and γ is the intrinsic Berry curvature contribution to the AHE (29–31). When we plot ρ AH against ρ x x 2 , we observe that a linear scaling appears to hold below ~230 K (Fig. 4B). It is possible that the deviation from linearity at high temperature arises from cancellations of Berry curvature associated with thermal broadening of the Fermi-Dirac distribution, as recently proposed for the AHE in metals (33). From the linear fit, we find that the intrinsic Berry curvature contribution to the AHE is γ = 870 ohm − 1 cm − 1 . This large intrinsic AHE leads us to consider the role of the Weyl lines in producing a large Berry curvature. To explore this question, we compare the intrinsic AHE measured in transport with a prediction based on ARPES and DFT. We observe in first-principles calculations that the Berry curvature distribution, calculated in the presence of SOC, is dominated by the topological node lines (Fig. 4C) (28). Next we investigate the Berry curvature up to a given binding energy to predict σ AH int as a function of the Fermi level. Then we set the Fermi level from ARPES, predicting σ AH int = 770 − 100 + 130 ohm − 1 cm − 1 (Fig. 4D). This finding is in good agreement with the value extracted from transport, suggesting that the topological node lines contribute substantially to the large AHE in Co 2 MnGa.

Fig. 4 Giant anomalous Hall transport and topological Weyl lines. (A) The Hall conductivity σ x y , measured as a function of applied magnetic field μ 0 H at several representative temperatures T, after two-point averaging of the raw data (fig. S5), with μ 0 H applied along [ 110 ] and current along [ 001 ] . (Inset) The anomalous Hall conductivity, σ AH ( T ) , obtained from σ x y . (B) The anomalous Hall resistivity ρ AH plotted against ρ x x 2 , for various values of T, as indicated by the colors: blue (2 K) → red (300 K). A linear scaling relation estimates the intrinsic Berry curvature contribution to the AHE, given by the slope of the line (29–31). Error bars indicate uncertainty in ρ AH estimated from the Hall resistivity (fig. S5). (C) (Bottom) z component of the Berry curvature, calculated with SOC, integrated up to E B = −0.09 eV, | Ω x y | . (Top) The ARPES constant-energy surface at the corresponding E B (same as Fig. 1D). The correspondence between ARPES and DFT suggests that the Berry curvature is dominated by the Weyl lines. (D) Prediction of σ AH int by integrating the Berry curvature from DFT up to a given E B (red curve), with E F set from ARPES (green shading indicates experimental error) and compared to the estimated σ AH int from transport. The prediction from ARPES and DFT is consistent with transport, suggesting that the Weyl fermion lines dominate the giant, intrinsic AHE in Co 2 MnGa.

Our ARPES and corresponding transport experiments, supported by ab initio calculation, provide evidence for magnetic Weyl fermion lines and drumhead topological surface states in the room temperature ferromagnet Co 2 MnGa. With 1651 magnetic space groups and thousands of magnets in 3D solids, the experimental methodology of transport-bulk-boundary exploration established in this work can be a valuable guideline in probing and discovering other topological phenomena on the surfaces and the bulk of magnetic or strongly correlated electronic materials.

Supplementary Materials science.sciencemag.org/content/365/6459/1278/suppl/DC1 Materials and Methods Supplementary Text Figs. S1 to S22 References (35–40)

http://www.sciencemag.org/about/science-licenses-journal-article-reuse This is an article distributed under the terms of the Science Journals Default License.