Abstract
The quantum behaviour of electrons in materials is the foundation of modern electronics and information technology^{1,2,3,4,5,6,7,8,9,10,11}, and quantum materials with topological electronic and optical properties are essential for realizing quantized electronic responses that can be used for next generation technology. Here we report the first observation of topological quantum properties of chiral crystals^{6,7} in the RhSi family. We find that this material class hosts a quantum phase of matter that exhibits nearly ideal topological surface properties originating from the crystals’ structural chirality. Electrons on the surface of these crystals show a highly unusual helicoid fermionic structure that spirals around two highsymmetry momenta, indicating electronic topological chirality. The existence of bulk multiply degenerate band fermions is guaranteed by the crystal symmetries; however, to determine the topological invariant or charge in these chiral crystals, it is essential to identify and study the helicoid topology of the arc states. The helicoid arcs that we observe on the surface characterize the topological charges of ±2, which arise from bulk higherspin chiral fermions. These topological conductors exhibit giant Fermi arcs of maximum length (π), which are orders of magnitude larger than those found in known chiral Weyl fermion semimetals^{5,8,9,10,11}. Our results demonstrate an electronic topological state of matter on structurally chiral crystals featuring helicoidarc quantum states. Such exotic multifold chiral fermion semimetal states could be used to detect a quantized photogalvanic optical response, the chiral magnetic effect and other optoelectronic phenomena predicted for this class of materials^{6}.
Main
The discovery of topological insulators has inspired the search for a wide variety of topological conductors^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26}. One example of a topological conductor is the Weyl semimetal, which features emergent Weyl fermions as lowenergy excitations of electrons. These Weyl fermions are associated with topological chiral charges (Chern numbers) that are located at twofolddegenerate band crossings^{1,4,5,8,9,11,12,13,14} in momentum space. Besides twofolddegenerate Weyl fermions^{4,5,8,9,10,11,12,13,14,15}, various types of emergent topological chiral fermion may appear, including Kramers–Weyl fermions^{6,7} and higherfold band fermions^{16,17,18}. Recently, a few noncentrosymmetric crystals were identified in which a band inversion gives rise to a Weyl semimetal state^{4,8,9,10,11,20,21,22,23}. However, all of these materials suffer from several drawbacks, such as large numbers of Weyl fermions, Weyl fermions close to each other in momentum space, and short Fermi arcs, which are much (orders of magnitude) less topologically robust than the ideal case allowed by theory. To thoroughly explore and utilize the robust and unusual quantum phenomena induced by chiral fermions in optics or magnetotransport, topological conductors with nearideal electronic properties or new types of topological conductor are needed^{1,7}.
A different approach towards searching for ideal topological conductors is to examine crystal symmetries, which can also lead to topological band crossings^{2,3,8}. For instance, it has been shown that nonsymmorphic symmetries can guarantee the existence of band crossings for certain electron fillings^{16,19,24}. As another example we might consider structurally chiral crystals, the lattices of which possess no inversion, mirror or rotoinversion symmetries^{6}. Structurally chiral crystals are expected to host a variety of topological band crossings, which are guaranteed to be pinned to timereversalinvariant momenta^{6,25}. Moreover, such crystals naturally give rise to a quantized circular photogalvanic current, the chiral magnetic effect and other novel transport and optics effects that are forbidden in known topological conductors, such as TaAs^{8,9,20,21,22,23}.
Incorporating these paradigms into a broader search, we have studied various candidate nonmagnetic and magnetic conductors, such as MoTe_{2}, WTe_{2}, Mn_{3}Ge, Ge_{3}Sn, Mn_{3}Sn, Na_{3}Bi, GdPtBi, the LuPtSb family, HgCr_{2}Se_{4}, pyrochlore iridates, LaPtBi, Co_{3}Sn_{2}S_{2}, Fe_{3}Sn_{2} and the XSi (X = Rh, Co) family, with advanced spectroscopic techniques. Many of these materials exhibit either large coexisting trivial bulk Fermi surfaces or surface reconstruction masking topological states. As often is the case with surfacesensitive techniques, the experimentally realized surface potential associated with a cleaved crystal may or may not allow these unusual electronic states to be observed. Thus, despite the new search paradigms, the discovery of topological materials suitable for spectroscopic experiments has remained a considerable challenge. Of all the materials that we explored, we observed that the XSi family of chiral crystals comes close to the experimental realization of the soughtafter ideal topological conductor. Additionally, it is a new type of topological conductor beyond Weyl semimetals. Here we report highresolution angleresolved photoemission spectroscopy (ARPES) measurements in combination with stateoftheart ab initio calculations to demonstrate new topological chiral properties in CoSi and RhSi. These chiral crystals approach ideal topological conductors because of their large Fermi arcs, which is related to the fact that they host the minimum nonzero number of chiral fermions—topological properties that we observe experimentally for the first time, to our knowledge.
The XSi family of materials crystallizes in a structurally chiral cubic lattice with space group P2_{1}3, number 198 (Fig. 1a, Extended Data Figs. 1, 2). We confirmed the chiral crystal structure of our CoSi samples using singlecrystal Xray diffraction (Fig. 1b, Extended Data Fig. 3, Extended Data Table 1) and the associated threedimensional Fourier map (Fig. 1c). We found a Flack factor of about 91%, which indicates that our samples are predominantly of a single structural chirality. Ab initio calculations of the electronic bulk band structure predict that both chiral crystals exhibit a threefold degeneracy at Γ near the Fermi level, E_{F} (Fig. 1d, Extended Data Fig. 4), which is described by a lowenergy Hamiltonian that exhibits a threefold fermion associated with Chern number +2 (refs ^{17,18}). We refer to this Chern number as chiral charge—a usage of the term ‘chiral’ that is distinct from the notion of structural chirality defined above and that motivates our use of the term ‘chiral fermion’ to describe these topological band crossings. The R point hosts a fourfold degeneracy corresponding to a fourfold fermion with Chern number −2. These two higherfold chiral fermions are pinned to opposite timereversalinvariant momenta and are consequently constrained to be maximally separated in momentum space (Fig. 1e), suggesting that CoSi might provide a nearideal platform for accessing topological phenomena using a variety of techniques. The hole pocket at M is topologically trivial at the band relevant for lowenergy physics, but it is well separated in momentum space from the Γ and R topological crossings, and it is not expected to affect topological transport, such as the chiral anomaly (we note that the longitudinal conductivity σ_{xx} does not correspond to topological transport). The two higherfold chiral fermions lead to a net Chern number of zero in the entire bulk Brillouin zone (BZ), as expected from theoretical considerations. CoSi and RhSi also satisfy a key criterion for an ideal topological conductor, namely, they have only two chiral fermions in the bulk BZ, the minimum nonzero number allowed^{6}.
The two chiral fermions remain topologically nontrivial over a wide energy range. In particular, CoSi maintains constantenergy surfaces with nonzero Chern number over an energy window of 0.85 eV, whereas for RhSi this window is 1.3 eV (Fig. 1d). This prediction suggests that XSi satisfies another criterion for an ideal topological conductor: a large topologically nontrivial energy window. A Fermi surface calculated ab initio shows that the projection of the higherfold chiral fermions to the (001) surface results in a hole (electron) pocket at \(\bar{{\rm{\Gamma }}}\left(\bar{{\rm{M}}}\right)\) with Chern number +2 (−2; Fig. 1f). As a result, we expect that XSi hosts Fermi arcs of length π spanning the entire surface BZ, again suggesting that these materials may realize a nearideal topological conductor.
Using lowphotonenergy ARPES, we experimentally study the (001) surface of CoSi and RhSi to reveal their surface electronic structure. For CoSi, the measured constantenergy contours show the following dominant features: two concentric contours around the \(\bar{{\rm{\Gamma }}}\) point, a faint contour at the \(\bar{{\rm{X}}}\) point and long winding states extending along the \(\bar{{\rm{M}}}\bar{{\rm{\Gamma }}}\bar{{\rm{M}}}\) direction (Fig. 2a). Both the \(\bar{{\rm{\Gamma }}}\) and \(\bar{{\rm{X}}}\) pockets show a holelike behaviour (Extended Data Fig. 5). The measured surface electronic structure for RhSi shows similar features (Fig. 2b, Extended Data Fig. 1). Using only our spectra, we first sketch the key features of the experimental Fermi surface for CoSi (Fig. 2c). Then, to better understand the kspace trajectory of the long winding states in CoSi, we study Lorentzian fits to the momentum distribution curves (MDCs) of the ARPES spectrum (Extended Data Fig. 6). We plot the Lorentzian peak positions as the extracted band dispersion (Fig. 2d, e) and we find that the long winding states extend from the centre of the BZ to the \(\bar{{\rm{M}}}\) pocket (Fig. 2f). To better understand the nature of these states, we analyse the photon energy dependence of the ARPES results and find that the long winding states do not disperse as we vary the photon energy, suggesting that they are surface states (Extended Data Fig. 7). Moreover, we observe an overall agreement between the ARPES data and the Fermi surface calculated ab initio, where topological Fermi arcs connect the \(\bar{{\rm{\Gamma }}}\) and \(\bar{{\rm{M}}}\) pockets (Fig. 1f). Taken together, these results suggest that the long winding states observed in ARPES may be topological Fermi arcs.
Grounded on the framework of topological band theory, the bulk–boundary correspondence of chiral fermions makes it possible to use ARPES (spectroscopic) measurements to determine the Chern numbers of a crystal by probing the surfacestate dispersion (Fig. 3a, Methods). Such spectroscopic methods to determine Chern numbers have become well accepted in the field^{26}. Using such an approach, we provide two spectral signatures of Fermi arcs in CoSi. We first look at the dispersion of the candidate Fermi arcs along a pair of energy–momentum cuts on opposite sides of the \(\bar{{\rm{\Gamma }}}\) pocket, taken at fixed momentum +k_{x} (cut I) and −k_{x} (cut II; Fig. 2f). In cut I, we observe two rightmoving chiral edge modes (Fig. 3b, c). Because the cut passes through two BZs (Fig. 2f), we associate one rightmoving mode with each BZ. Next, we fit Lorentzian peaks to the MDCs and we find that the extracted dispersion again suggests two chiral edge modes (Fig. 3d). Along cut II, we observe two leftmoving chiral edge modes (Fig. 3e, f). Consequently, one chiral edge mode is observed for each measured surface BZ on cuts I and II, but with opposite Fermi velocity directions. In this way, our ARPES spectra suggest that the number of chiral edge modes n changes by +2 when the kslice is swept from cut I to cut II. This again suggests that the long winding states are topological Fermi arcs. Moreover, these ARPES results imply that projected topological charge with net Chern number +2 resides near \(\bar{{\rm{\Gamma }}}\).
Next we search for other Chern numbers encoded by the surfacestate band structure. We study an ARPES energy–momentum cut on a loop \({\mathscr{P}}\) enclosing \(\bar{{\rm{M}}}\) (Fig. 4a, inset). Again following the bulk–boundary correspondence, we aim to extract the Chern number of chiral fermions projecting on \(\bar{{\rm{M}}}\). The cut \({\mathscr{P}}\) shows two rightmoving chiral edge modes dispersing towards E_{F} (Fig. 4a, b), suggesting a Chern number of −2 on the associated bulk manifold. Furthermore, the surface spectral weight along \({\mathscr{P}}\) calculated ab initio is consistent with our experimental results (Fig. 4c). Our ARPES spectra on cuts I, II and \({\mathscr{P}}\) suggest that CoSi hosts a projected chiral charge of +2 at \(\bar{{\rm{\Gamma }}}\) with its partner chiral charge of −2 projecting on \(\bar{{\rm{M}}}\). This again provides evidence that the long winding states are a pair of topological Fermi arcs that traverse the surface BZ on a diagonal, connecting the \(\bar{{\rm{\Gamma }}}\) and \(\bar{{\rm{M}}}\) pockets. Our ARPES spectra of RhSi also provide evidence for gigantic topological Fermi arcs following a similar analysis (Extended Data Fig. 1).
To further explore the topological properties of CoSi, we examine in greater detail the structure of the Fermi arcs near \(\bar{{\rm{M}}}\). We consider the dispersion on \({\mathscr{P}}\) (plotted as a magenta loop in Fig. 4d, inset) and we also extract the dispersion from Lorentzian fitting on a second, tighter circle (black loop; Extended Data Fig. 8). We observe that as we decrease the binding energy (approaching E_{F}), the extracted dispersion spirals clockwise on both loops, suggesting that as a given k point traverses the loop, the energy of the state does not return to its initial value after a full cycle. Such a nontrivial electronic dispersion directly signals a projected chiral charge at \(\bar{{\rm{M}}}\) (Fig. 4d). In fact, the extracted dispersion is characteristic of the helicoid structure of topological Fermi arcs as they wind around a chiral fermion (Fig. 4e), suggesting that CoSi and RhSi^{17} are rare examples of noncompact surfaces in nature^{6,27}.
To further understand these experimental results, we consider the spectral weight for the (001) surface calculated ab initio, and we observe a pair of Fermi arcs winding around the \(\bar{{\rm{\Gamma }}}\) and \(\bar{{\rm{M}}}\) pockets counterclockwise and clockwise, respectively, with decreasing binding energy (approaching E_{F}; Fig. 4f). The clockwise winding around \(\bar{{\rm{M}}}\) is consistent with our ARPES observation of a −2 projected chiral charge. Moreover, from our ab initio calculations, we predict that the −2 charge projecting to \(\bar{{\rm{M}}}\) arises from a fourfold chiral fermion at the bulk R point (Fig. 1d). The +2 chiral charge, which we associate with \(\bar{{\rm{\Gamma }}}\) from ARPES (Fig. 3), is further consistent with the threefold chiral fermion predicted at the bulk Γ point. By fully accounting for the predicted topological charges in the experiment, our ARPES results suggest the manifestation of a topological chiral crystal in CoSi. We can similarly account for the predicted topological charges of multifold fermions in RhSi (Extended Data Figs. 1, 2).
The surfacestate dispersions in our ARPES spectra, taken together with the topological bulk–boundary correspondence, demonstrate that CoSi and RhSi are topological chiral crystals^{6,7}. This experimental result is further consistent with the numerical result of firstprinciples calculations of the surfacestate dispersions and topological invariants. Unlike previously reported Weyl semimetals, the Fermi arcs that we observe in CoSi and RhSi stretch diagonally across the entire (001) surface BZ, from \(\bar{{\rm{\Gamma }}}\) to \(\bar{{\rm{M}}}\). In fact, the Fermi arcs in XSi are longer than those found in TaAs by a factor of thirty. Our surface band structure measurements also demonstrate two wellseparated Fermi pockets carrying Chern number ±2. Lastly, for the first time to our knowledge, we observe in an electronic material the helicoid structure of topological Fermi arcs. Our results suggest that CoSi and RhSi are excellent candidates for studying topological phenomena distinct to chiral fermions by using a variety of techniques^{4,8,11}.
Crucially for applications, the topologically nontrivial energy window in CoSi and RhSi is an order of magnitude larger than that in TaAs^{6,21,22}, rendering its quantum properties robust against changes in surface chemical potential and disorder. Moreover, the energy offset between the higherfold chiral fermions at Γ and R is predicted to be about 225 meV. Such an energy offset is essential for inducing the chiral magnetic effect and its optical analogue^{28}, as well as the photogalvanic effect (optical)^{6,17,29}, and our results on RhSi samples suggest that it can be used to observe a fourfold quantized circular photogalvanic current^{17}. When coupled to a compatible superconductor, CoSi and RhSi are compelling platforms for studying the superconducting pairing of Fermi surfaces with nonzero Chern numbers, which may be promising for realizing a new type of topological superconducting phase proposed recently^{17,30}, which can be probed with scanning tunnelling microscopybased spectroscopy^{31}. CoSi and RhSi further open the door to the exploration of other exotic quantum phenomena when combined with the isochemical material FeSi. Fe_{1−x}Co_{x}Si may simultaneously host kspace (chiral fermions) and realspace (skyrmions) topological defects and their interplay, which can also be probed by scanning tunnelling microscopy/spectroscopy^{31}. Through our observation of a helicoid surface state and its topological properties, our results suggest the experimental discovery of the first structurally chiral crystal that is also topological. In this way, our work provides a nextgeneration platform that could be used in the further study of, and the search for, new types of topological chiral crystal^{6}.
While the present article was under review, a related work studying bulk dispersion in CoSi was reported^{32}.
Methods
Observation of topological Fermi arcs in RhSi
Motivated by an interest in establishing families of Kramers–Weyl and closely related topological materials^{9,34,35,36,37,38,39,40,41,42,43,44,45}, we expand our experiment to include rhodium silicide, RhSi, an isoelectronic cousin of CoSi. RhSi crystallizes in a chiral cubic lattice, space group P2_{1}3, number 198. The calculated electronic band structure is generally similar to that of CoSi (sidebyside comparison in Fig. 1d). Constantenergy contours measured by ARPES further suggest features similar to those of CoSi (Extended Data Fig. 1a). In particular, we observe contours at the \(\bar{{\rm{\Gamma }}}\), \(\bar{{\rm{X}}}\) and \(\bar{{\rm{Y}}}\) points and long winding states that extend diagonally from \(\bar{{\rm{\Gamma }}}\) to \(\bar{{\rm{M}}}\). Again taking advantage of the bulk–boundary correspondence, we focus on counting chiral edge modes to determine the topological nature of RhSi (Extended Data Fig. 1b, c). A secondderivative plot of the Fermi surface further suggests the presence of long states stretching diagonally across the BZ, motivating a study of energy–momentum cuts through these states (Extended Data Fig. 1d–g). The cuts show that the long states take the form of a rightmoving chiral edge mode (cut I) and a leftmoving chiral edge mode (cut II) on opposite sides of \(\bar{{\rm{\Gamma }}}\). The net difference in the number of rightmoving chiral edge modes suggests that a Chern number of +2 projects to \(\bar{{\rm{\Gamma }}}\) and that the long states are topological Fermi arcs. Proceeding again by analogy to CoSi, we study the band structure along a loop \({\mathscr{M}}\) enclosing \(\bar{{\rm{M}}}\), and we observe two rightmoving chiral edge modes, suggesting a Chern number of −2 at R. These results suggest that RhSi, an isoelectronic cousin of CoSi, is another example of a nearideal topological conductor.
Helicoid structure of the arcs and topology in RhSi
We study the Fermi arcs near \(\bar{{\rm{M}}}\) in RhSi, by analogy with the analysis performed for CoSi (Fig. 4). First, we consider ARPES energy–momentum cuts on an inner and outer loop enclosing \(\bar{{\rm{M}}}\), and we observe signatures of two rightmoving chiral edge modes on each loop, suggesting an enclosed projected chiral charge of −2 (Extended Data Fig. 2a–d). We further fit Lorentzians to the MDCs and we find that the extracted dispersion exhibits clockwise winding on both loops for decreasing binding energy (approaching E_{F}; Extended Data Fig. 2e, f). This dispersion again suggests a chiral charge −2 at \(\bar{{\rm{M}}}\) and is characteristic of the helicoid structure of Fermi arcs as they wind around a chiral fermion. Lastly, we perform ab initio calculations of the constantenergy contours (Extended Data Fig. 2g), which exhibit the same winding pattern. These ARPES results suggest that RhSi, like CoSi, exhibits helicoid topological Fermi arcs.
Bulk–boundary correspondence in CoSi and RhSi
By measuring the surfacestate band structure of a crystal, ARPES is capable of demonstrating Fermi arcs and counting Chern numbers^{33}. We briefly review the details of this method. First, recall that topological invariants are typically defined for a gapped system. A topological conductor is by definition gapless so we cannot define a topological invariant for the full bulk BZ of the threedimensional system. However, we may be able to choose certain twodimensional kspace manifolds where the band structure is fully gapped, and a twodimensional invariant can then be defined on this slice of momentum space (Fig. 3a). We consider specifically the case of a chiral fermion in a topological conductor and we study a gapped momentumspace slice that cuts in between two chiral fermions. By definition, a chiral fermion is associated with a nonzero Chern number, so at least some of these slices will be characterized by a nonzero Chern number. Following the bulk–boundary correspondence, these slices will then contribute chiral edge modes to the surfacestate dispersion. If we image by collecting together the chiral edge modes from all of the gapped slices, we will assemble the entire topological Fermi arc of the topological conductor. If we run the bulk–boundary correspondence in reverse, we can instead measure the surface states by ARPES and count the chiral edge modes on a particular onedimensional slice of the surface BZ to determine the Chern number of the underlying twodimensional slice of the bulk BZ. These Chern numbers in turn fix the chiral charges of the topological fermions.
Next we highlight two spectral signatures that can determine the chiral charges specifically in XSi (X = Co, Rh). First, we consider chiral edge modes along straight kslices. Theoretically, the twodimensional kslices on the two sides of the chiral fermion at Γ are related by timereversal symmetry and therefore should have equal and opposite Chern numbers (n_{l} = −n_{r}; Fig. 3b). On the other hand, because the threefold chiral fermion at Γ is predicted to carry chiral charge +2, we expect that the difference should be n_{l} − n_{r} = +2, resulting in n_{l(r)} = ±1. Therefore we expect one net leftmoving chiral edge mode on one cut and one net rightmoving chiral edge mode on the other. For the second signature, we study the chiral edge modes along a closed loop in the surface BZ. Any loop that encloses the projected chiral charge of +2 at \(\bar{{\rm{\Gamma }}}\) or −2 at \(\bar{{\rm{M}}}\) has this number of net chiral edge modes along this path. By taking advantage of this correspondence, we can count surface states on loops in our ARPES spectra to determine enclosed projected chiral charges.
Growth of CoSi and RhSi single crystals
Single crystals of CoSi were grown using a chemical vapour transport technique. First, polycrystalline CoSi was prepared by arcmelting stoichiometric amounts of Co slices and Si pieces, which were crushed and ground into a powder. The sample was then sealed in an evacuated silica tube with an iodine concentration of approximately 0.25 mg cm^{−3}. The transport reaction took place at a temperature gradient from 1,000 °C (source) to 1,100 °C (sink) for two weeks. The resulting CoSi single crystals had a metallic lustre and varied in size from 1 mm to 2 mm. Single crystals of RhSi were grown from a melt using the vertical Bridgman crystal growth technique at a nonstoichiometric composition. In particular, we induced a slight excess of Si to ensure flux growth inside the Bridgman ampoule. First, a polycrystalline ingot was prepared by premelting the highly pure constituents under an argon atmosphere using an arc furnace. The crushed powder was poured into a customdesigned sharpedged alumina tube and then sealed inside a tantalum tube under an argon atmosphere. The sample was heated to 1,550 °C and then slowly pooled to the cold zone at a rate of 0.8 mm h^{−1}. Single crystals on average about 15 mm in length and about 6 mm in diameter were obtained.
Xray diffraction
Single crystals of CoSi were mounted on the tips of Kapton loops. The lowtemperature (100 K) intensity data were collected on a Bruker Apex II Xray diffractometer with Mo radiation Kα_{1} (λ = 0.71073 Å^{−1}). Measurements were performed over a full sphere in kspace with 0.5° scans in angle ω with an exposure time of 10 s per frame (Extended Data Fig. 3). The SMART software was used for data acquisition. The extracted intensities were corrected for Lorentz and polarization effects with the SAINT program. Numerical absorption corrections were accomplished with XPREP, which is based on faceindexed absorption^{46}. The twin unit cell was tested. With the SHELXTL package, the crystal structures were solved using direct methods and refined by fullmatrix leastsquares on F2 (ref. ^{47}) No vacancies were observed according to the refinement and no residual electron density was detected, indicating that the CoSi crystals were of high quality.
Sample surface preparation for ARPES
For CoSi single crystals, the surface preparation procedure followed the conventional in situ mechanical cleaving approach. This cleaving method resulted in a low success rate and the resulting surface typically appeared rough under a microscope. We speculate that the difficulty in cleaving CoSi single crystals may be a result of its cubic structure, strong covalent bonding and lack of a preferred cleaving plane. For RhSi single crystals, their large size enabled mechanical cutting and polishing along the (001) surface. An in situ sputtering and annealing procedure, combined with lowenergy/reflection highenergy electron diffraction characterization, was used to obtain a clean surface suitable for ARPES measurements. The typical spectral line width was narrowed and the background signal was reduced for these samples compared with the mechanically cleaved CoSi samples.
ARPES
ARPES measurements were carried out at beamlines 10.0.1 and 4.0.3 at the Advanced Light Source in Berkeley, California, USA. A Scienta R4000 electron analyser was used at beamline10.0.1 and a Scienta R8000 was used at beamline 4.0.3. At both beamlines the angular resolution was <0.2° and the energy resolution was better than 20 meV. Samples were cleaved or sputtered/annealed in situ and measured under vacuum better than 5 × 10^{−11} torr at T < 20 K.
Firstprinciples calculations
Numerical calculations of XSi (X = Co, Rh) were performed within the density functional theory framework using the OPENMX package and the full potential augmented planewave method as implemented in the package WIEN2k^{48,49,50}. The generalized gradient approximation was used^{51}. Experimentally measured lattice constants were used in the density functional theory calculations of material band structures^{52}. A Γcentred kpoint 10 × 10 × 10 mesh was used and spin–orbit coupling was included in selfconsistent cycles. To generate the (001) surface states of CoSi and RhSi, Wannier functions were generated using the p orbitals of Si and the d orbitals of Co and Rh. The surface states were calculated for a semiinfinite slab by the iterative Green’s function method.
Electronic bulk band structure
The electronic bulk band structure of CoSi is shown with and without spin–orbit coupling in Extended Data Fig. 4. The small energy splitting that arises from introducing spin–orbit coupling is approximately 40 meV, which is negligible compared to the about 1.2 eV topologically nontrivial energy window. Our results suggest that we do not need to consider spin–orbit coupling, either theoretically or experimentally, to demonstrate a chiral charge in CoSi.
Tracking the Fermi arcs by fitting ARPES MDCs
The following procedure was performed to track the Fermi arcs in the surface BZ of CoSi (Extended Data Fig. 6a). At E = E_{F}, MDCs were collected for various fixed k_{y} values (extending along the region of interest). Each MDC was fitted with a Lorentzian function to pinpoint the k_{x} value that corresponds to the peak maximum. The peak maximum of the fitted MDC is plotted on top of the measured Fermi surface and marked with blue circles in Extended Data Fig. 6b. Where necessary, the peak maximum corresponding to the \(\bar{{\rm{X}}}\) pockets is annotated on each MDC. For the chiral edge modes discussed in the main text and shown in the Extended Data, a similar MDC fitting procedure was used at different binding energies to track the dispersion.
Study of Fermi arcs with varying photon energy
Extended Data Fig. 7 shows an MDC cutting through the Fermi arc as a function of incident photon energy. Using methods established previously^{8} we find that within the experimental resolution the Fermi arc shows negligible variation from 80 eV to 110 eV, providing additional evidence that it is indeed a surface state.
Data availability
The data supporting the findings of this study are available within the paper, and other findings of this study are available from the corresponding author upon reasonable request.
Additional information
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Acknowledgements
Work at Princeton and Princetonled ARPES measurements and theoretical work reported here were supported by the US Department of Energy under a Basic Energy Sciences grant (number DOE/BES DEFG0205ER46200). M.Z.H. acknowledges a Visiting Scientistship at the Materials Science Division of Lawrence Berkeley National Laboratory. This research used resources of the Advanced Light Source, which is a DOE Office of Science Facility, under contract number DEAC0205CH11231. S.J. was supported by the National Natural Science Foundation of China (U1832214, 11774007), the National Key R&D Program of China (2018YFA0305601) and the Key Research Program of the Chinese Academy of Science (grant number XDPB081). H.L. acknowledges Academia Sinica, Taiwan for support under the Innovative Materials and Analysis Technology Exploration (ASiMATE10711) programme. T.R.C. is supported by the Young Scholar Fellowship Program of the Ministry of Science and Technology (MOST) in Taiwan, under the MOST Grant for the Columbus Program (MOST1072636M006004), National Cheng Kung University, Taiwan, and National Center for Theoretical Sciences (NCTS), Taiwan. C.F. acknowledges financial support by the ERC through Advanced Grant number 291472 (Idea Heusler) and 742068 (TOPMAT). The authors thank S. K. Mo and J. D. Denlinger for beamline support at the Advanced Light Source in Berkeley, California. We thank R. Dhall of the Molecular Foundry at LBL in Berkeley, California for assistance. M.Z.H. acknowledges support from the Miller Institute of Basic Research in Science at the University of California in Berkeley in the form of a Visiting Miller Professorship during the early stages of this work. We also thank G. Bian and H. Verlinde for discussions.
Author information
Author notes
These authors contributed equally: Daniel S. Sanchez, Ilya Belopolski, Tyler A. Cochran
Affiliations
Laboratory for Topological Quantum Matter and Advanced Spectroscopy (B7), Department of Physics, Princeton University, Princeton, NJ, USA
 Daniel S. Sanchez
 , Ilya Belopolski
 , Tyler A. Cochran
 , JiaXin Yin
 , Guoqing Chang
 , Nasser Alidoust
 , Daniel Multer
 , Songtian S. Zhang
 , Nana Shumiya
 , SuYang Xu
 & M. Zahid Hasan
International Center for Quantum Materials, School of Physics, Peking University, Beijing, China
 Xitong Xu
 , Xirui Wang
 , GuangQiang Wang
 & Shuang Jia
Department of Chemistry, Louisiana State University, Baton Rouge, LA, USA
 Weiwei Xie
Max Planck Institute for Chemical Physics of Solids, Dresden, Germany
 Kaustuv Manna
 , Vicky Süß
 & Claudia Felser
Institute of Physics, Academia Sinica, Taipei, Taiwan
 ChengYi Huang
 & Hsin Lin
Rigetti Quantum Computing, Berkeley, CA, USA
 Nasser Alidoust
Department of Chemistry, Princeton University, Princeton, NJ, USA
 Daniel Multer
Department of Physics, National Cheng Kung University, Tainan, Taiwan
 TayRong Chang
Collaborative Innovation Center of Quantum Matter, Beijing, China
 Shuang Jia
CAS Center for Excellence in Topological Quantum Computation, University of the Chinese Academy of Science, Beijing, China
 Shuang Jia
Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA
 M. Zahid Hasan
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Contributions
D.S.S. and M.Z.H. conceived the project. D.S.S., I.B., T.A.C. and N.A. conducted the ARPES experiments in consultation with S.Y.X. and M.Z.H.; X.X., W.X., X.W., G.Q.W. and S.J. synthesized and characterized the RhSi and CoSi samples; K.M., V.S. and C.F. synthesized the RhSi samples; G.C., C.Y.H., T.R.C. and H.L. performed the firstprinciples/density functional theory calculations; D.S.S., I.B. and T.A.C. performed the analysis, interpretation and figure development in consultation with J.X.Y., G.C., N.A., D.M., S.S.Z., N.S., S.Y.X. and M.Z.H.; D.S.S. wrote the manuscript in consultation with I.B., J.X.Y., G.C., S.Y.X. and M.Z.H.; M.Z.H. supervised the project.
Competing interests
The authors declare no competing interests.
Corresponding author
Correspondence to M. Zahid Hasan.
Extended data figures and tables
Extended Data Fig. 1 Topological arcs in the structurally chiral crystal RhSi.
a, ARPESmeasured Fermi surface and constantbindingenergy contours with an incident photon energy of 82 eV at 10 K. The BZ boundary is marked in blue. b, Threedimensional bulk and twodimensional surface BZ with higherfold chiral fermions (±). The planes outlined in blue and the red cylinder are twodimensional manifolds with indicated Chern number^{33} n. The cylinder enclosing the bulk chiral fermion at Γ has n = +2. c, Fermi arcs (orange) connect the projected chiral fermions. Energy dispersion cuts show that these two chiral edge modes are timereversed partners propagating in opposite directions. d, Secondderivative Fermi surface with the straight and loop cuts of interest marked. e, ARPES spectrum along a loop \({\mathscr{M}}\), showing two rightmoving chiral edge modes, suggesting that the fourfold chiral fermion at R carries Chern number −2. f, ARPES spectrum along cut I, showing a rightmoving chiral edge mode. g, ARPES spectrum along cut II on the opposite side of the threefold chiral fermion at \(\bar{{\rm{\Gamma }}}\) (illustrated by the blue sphere), showing a leftmoving chiral edge mode.
Extended Data Fig. 2 Helicoid arcs in RhSi.
a, Energy dispersion cut on an inner loop of radius 0.18π/a enclosing \(\bar{{\rm{M}}}\). b, Lorentzian fits (red traces) to the MDCs (blue dots) to track the observed chiral edge modes. c, d, Similar analysis to a and b, but for an outer loop of radius 0.23π/a. Black arrows show the Fermi velocity direction for the chiral edge modes. e, f, Top (e) and perspective (f) view of the helicoid dispersion extracted from the MDCs, plotted on two concentric loops, suggesting a clockwise spiral with decreasing binding energy. g, Constantenergy contours calculated ab initio, showing a consistent helicoid structure.
Extended Data Fig. 3 Xray diffraction study.
Singlecrystal Xray diffraction precession image of the (0kl) planes in the reciprocal lattice of CoSi. The resolved spots from scans I and III are consistent with space group P2_{1}3 at 100 K. This is also shown in the reflection intersection, scan II.
Extended Data Fig. 4 Electronic bulk band topology.
a, Band structure of CoSi in the absence of spin–orbit coupling interactions. b, Band structure in the presence of spin–orbit coupling. The highest valence and lowest conduction bands are coloured in blue and red, respectively.
Extended Data Fig. 5 Surveying the \(\bar{{\rm{\Gamma }}}\) and \(\bar{{\rm{X}}}\) pockets on the (001) surface of CoSi.
a, ARPESmeasured Fermi surface with the BZ boundary (red dashed line) and the location of the energy–momentum cut of interest (magenta dashed line). b, \(\bar{{\rm{\Gamma }}}\bar{{\rm{X}}}\bar{{\rm{\Gamma }}}\) highsymmetryline energy–momentum cut. c, Secondderivative plot of b. We observe an outer and an inner holelike band at \(\bar{{\rm{\Gamma }}}\), whereas at \(\bar{{\rm{X}}}\) we observe signatures of a single holelike band.
Extended Data Fig. 6 Tracking the Fermi arcs by fitting Lorentzians to MDCs.
a, Zoomedin region of the Fermi surface (Fig. 2d), with the Fermi arcs tracked (blue circles) and the surface BZ marked (red dashed lines). b, Representative fits of Lorentzian functions (red lines) to the MDCs (filled blue circles). The peaks indicate the extracted positions of the Fermi arc (open blue circles) and the \(\bar{{\rm{X}}}\) pocket.
Extended Data Fig. 7 Photonenergy dependence of the Fermi arc in CoSi.
a, Location of the inplane momentum direction (black arrow) along which the photonenergy dependence was studied, plotted on the Fermi surface (Fig. 2d) with the surface BZ (red dotted lines). b, MDCs at E_{F} along the inplane momentum direction illustrated in a, obtained for a series of photon energies from 80 eV to 110 eV in steps of 2 eV. The peak associated with the Fermi arc shows negligible variation in photon energy (dashed line) within the experimental resolution, providing further evidence that the observed Fermi arc is indeed a surface state.
Extended Data Fig. 8 Systematics for the helicoid fitting in CoSi.
a, Fermi arc trajectory extracted from Lorentzian fits to the MDCs (blue open circles) near \(\bar{{\rm{M}}}\), with overlaid schematic of the observed features (orange lines). There are two closed contours enclosing \(\bar{{\rm{M}}}\) on which we count chiral edge modes: the outer loop (magenta, Fig. 4) and the inner loop (black). b, Energy–momentum cut along the inner loop (radius 0.14π/a), starting from the green notch in a and winding clockwise (black arrow in a). c, Lorentzian fits (red curves) to the MDCs (blue circles) to extract the helicoid dispersion of the Fermi arcs. We observe two rightmoving chiral edge modes dispersing towards E_{F} (dashed orange lines). The corresponding bulk manifold has Chern number^{13} n = −2.
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Topological triply degenerate point with double Fermi arcs
Nature Physics (2019)
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