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Chiral fermion reversal in chiral crystals

Results.Crystal structures and band calculations of RhSn and CoSi.Our single-crystal X-ray diffraction (XRD) measurements determine that both CoSi and RhSn crystalize in space group P213 (#198), whose lattices are chiral without inversion, mirror and roto-inversion symmetries. The lattice parameters in the refinement are a = 4.4445 Å, uCo = 0.1066 and uSi = 0.4064 for CoSi, and a = 5.1315 Å, uRh = 0.3557 and uSn = 0.6599 for RhSn. Detailed XRD data are listed in the Supplementary.cif files of RhSn and CoSi. The results show that they have opposite chirality in the lattices. In the view of [111] direction in Fig. 1a, b, the Co and Si atoms in CoSi form right- and left-handed helices, respectively. Instead, the Rh and Sn atoms in RhSn form left- and right-handed helices, respectively.Fig. 1Reversal of chiral lattices and chiral fermions between CoSi and RhSn. a, b Crystal structures of CoSi (a) and RhSn (b) in the view of [111] direction. The transparency of the balls indicates the depth of the atomic positions from top to the bottom. The red and blue arrows mark the handedness of the Co/Rh and Si/Sn helixes, respectively. c Bulk BZ and (001) and (110) surface BZs. The blue and red dots represent the locations of the spin-1 and charge-2 fermions, respectively. d Calculated FSs of RhSn without SOC in the bulk BZ. e WCCs of valence bands around Γ computed from (r cos θ, r sin θ, 0) to (r cos θ, r sin θ, 2π/c) with r = 0.084 Å−1. The evolution of WCCs for CoSi and RhSn is plotted as a function of θ in f and g, respectively. h Calculated bulk band structure of RhSn along the high-symmetry lines without SOC. The numbers mark the four bands related to the chiral fermions. i Calculated (001)-surface states of RhSn without SOC in four surface BZs. The yellow dashed circles and arrows indicate the momentum paths and directions of loops #1 and #2. j, k Surface band structures along loops #1 (j) and #2 (k) in i.Full size image First-principles calculations show that the bulk band structure of RhSn is similar to that of CoSi28,29,30. When spin–orbit coupling (SOC) is not included, the band structure in Fig. 1h shows a threefold-degenerate point at the Brillouin zone (BZ) centre Γ and a fourfold-degenerate point at the BZ corner R near the Fermi level (EF), where the quasiparticle excitations are named spin-1 and charge-2 fermions, respectively. The outer Fermi surfaces (FSs) in Fig. 1d are formed by bands 2 and 3 in Fig. 1h. We define bands 2 and 3 as valence and conduction bands, respectively, to calculate the chiral charges of these fermions. The Wannier charge centre (WCC) calculations of occupied states (Fig. 1e–g) indicate that the spin-1 fermions at Γ of CoSi and RhSn carry opposite chiral charges +2 and −2, respectively. According to the “no-go theorem”, the charge-2 fermions at R of CoSi and RhSn must carry the chiral charges −2 and +2, respectively. The opposite structural chirality does not affect the bulk band structures, but it changes the signs of the chiral charges of the degenerate points.It has been known that the surface projections of chiral fermions are surrounded by helical surface states35. The helical surface states for CoSi have been theoretically predicted28,30 and experimentally confirmed on the (001) surface31,33. Our calculations of the (001) surface states of RhSn in Fig. 1i–k also show helical surface states with chiral bands on the loops around the projections of the chiral fermions at Γ and R. The signs of the chiral charges dictate the handedness of the helical surface states, which can be directly detected by ARPES experiments.It should be mentioned that the SOC effects exist in these materials. Due to the lack of inversion symmetry, the bands of both bulk and surface states split when SOC is included, as shown in our calculations with SOC in the Supplementary Figs. 1 and 2. Since the splitting is not resolved in our ARPES data, for simplicity, our discussion is limited in the framework without SOC, as in the previous studies of CoSi and RhSi31,32,33.Soft X-ray ARPES results on the bulk states of RhSn.In the previous study, we have obtained flat (001) surfaces of CoSi single crystals by sputtering and annealing, and observed clear band dispersions of the bulk and surface states31. We applied this method to RhSn single crystals and obtained flat (001) and (110) surfaces. On both surfaces, the spin-1 and charge-2 fermions are projected to the surface BZ centre and boundary, respectively (Fig. 1c). It is expected to observe topological surface states on both surfaces. By analysing the handedness of the surface bands, we can determine experimentally whether the chiral fermions in these two materials are reversed.In ARPES experiments, photoelectrons excited by soft X-ray have much longer escape depth than those excited by vacuum ultraviolet (VUV) light. We thus can selectively probe the bulk and surface states using soft X-ray and VUV light. In Fig. 2, we summarise the ARPES data of RhSn collected with soft X-ray. Figure 2a–c shows the intensity maps at EF measured on the (001) surface with photon energies hν = 344 and 390 eV and on the (110) surface with hν = 395 eV. They are generally consistent with the calculated FSs in the kz = 0 and π planes and the Γ-X-R-M plane, respectively, in Fig. 2d–f.Fig. 2FSs and band structures of bulk states of RhSn. a–c ARPES intensity maps at EF showing the FSs in three high-symmetry planes #1 (a), #2 (b) and #3 (c). d–f Calculated FSs in planes #1 (d), #2 (e) and #3 (f). The dashed red lines in a–f indicate the bulk BZ boundary. g Locations of planes #1, #2 and #3 in the bulk BZ. h, i Curvature intensity maps of the ARPES data along M-Γ-M (h) and R-M-R (i). For comparison, we plot the corresponding calculated bands as red dashed curves on top of the experimental data in h and i. The ARPES data in a, b, c, h and i were collected with hν = 344, 390, 395, 435 and 490 eV, respectively.Full size image While the main features in the calculations are captured in the experimental data, there are some differences in detail. For example, the calculations in Fig. 2d show two intersecting elliptical FSs around M in the kz = 0 plane, whereas only one is observed in the experiments in Fig. 2a. Similarly, only the inner FS around M is observed in the Γ-X-R-M plane in Fig. 2c. The differences around M are due to the matrix element effects. Moreover, in the Γ-X-R-M plane, the experimental FS structure near Γ in Fig. 2c appears to be different from the calculations in Fig. 2f. According to the calculated band structure in Fig. 1h, band 2 forms two nearly rectangular FSs on both sides of Γ in the Γ-X-R-M plane in Fig. 2f. We only observe the rectangular FS on the left of Γ in Fig. 2c due to the matrix element effects. In addition, one can see four petal-like features around Γ in Fig. 2c, which derive from band 1 just below EF along Γ-R, as shown in Fig. 1h. One can see similar features near Γ with lower intensities in Fig. 2f, as the calculated contours at EF are the integral of spectral functions in an energy range with respect to EF.Figure 2h, i shows the band dispersions measured along M-R and Γ-M. For comparison, we plot the corresponding calculated bands as dashed curves on top of the experimental data. Most of the experimental band dispersions are well consistent with the calculations, except for slight shifts in energy. The experimental bands near Γ in Fig. 2i deviate obviously from band 1 along Γ-M in the calculations, probably due to the momentum broadening in the direction normal to the sample surface. The band dispersions near the calculated degenerate points at Γ and R are not very clear, making it difficult to directly identify the degenerate points in the experimental data. Nevertheless, the overall agreement in the FSs and band dispersions between experiments and calculations in Fig. 2 strongly supports the validity of the calculations, which indicate the existence of degenerate points at Γ and R in the bulk states of RhSn.VUV ARPES results on the surface states of RhSn.To clarify the topological properties of these degenerate points, we have investigated the electronic structures on the (001) and (110) surfaces using VUV ARPES. While the FSs measured by soft X-ray are located around the high-symmetry points in Fig. 2, we observe some extra features in between the projected bulk FSs in the VUV ARPES data in Fig. 3. Figure 3a–c shows that the extra features are present in the intensity maps at EF measured on the (001) surface at three different photon energies hν = 75, 90 and 60 eV. Their positions in the surface BZ do not change with photon energy (Fig. 3d). Moreover, the extra features only have a π-rotation symmetry about the time-reversal-invariant momenta. This is consistent with the fact that all crystalline symmetries except the in-plane translations in the bulk are broken at the surface and only time-reversal symmetry is left invariant36. Therefore, we determine that the extra features are of surface origin.Fig. 3(001) and (110) surface states of RhSn. a–c ARPES intensity maps at EF measured on the (001) surface at hν = 75 eV (a), 60 eV (b) and 90 eV (c). d Projected bulk FSs without SOC on the (001) surface. The yellow dashed curves are a guide to the surface Fermi arcs. The red crosses, green diamonds and black circles are extracted from peak positions of momentum distribution curves (MDCs) of the ARPES data in a, b and c, respectively. e Curvature intensity map of the ARPES data in a. The red dashed curve indicates one representative surface Fermi arc extending from \({\bar{\mathrm{\Gamma }}}\) to \({\bar{\mathrm{M}}}\) . f MDCs of the ARPES data in a. The blue curves indicate the MDCs along the high-symmetry lines. The black dots mark the high-symmetry points. The back dashed curve is a guide to the peak positions of the MDCs. g ARPES intensity map at EF measured on the (110) surface at hν = 75 eV. h Projected bulk FSs without SOC on the (110) surface.Full size image As seen in Fig. 3e, the surface states are arc-like and connect the projected FSs around \({\bar{\mathrm{\Gamma }}}\) and \({\bar{\mathrm{M}}}\) , which enclose the spin-1 and charge-2 fermions, respectively. There are two Fermi arcs emanating from each of the projected FSs, in agreement with the chiral charges ±2 in the calculations without SOC. It is noteworthy that the Fermi arcs connect the next-nearest neighbours of the projections, spanning one and a half surface BZs (~ 2 Å−1), which is the longest Fermi arc observed so far. The connection pattern is different from the calculations in Fig. 1i, in which the Fermi arcs connect the nearest neighbours of the projections. The surface used in the calculations should be different from the real surface in the experiments. The band structures of surface states are sensitive to specific surface configurations, but the topological properties of the surface states must not be changed.On the (110) surface, the spin-1 and charge-2 fermions are projected to the \({\tilde{\mathrm{\Gamma }}}\) and \({\tilde{\mathrm{R}}}\) points, respectively (Fig. 1c). We also observed two arc-like features emanating from the projected FS around \({\tilde{\mathrm{R}}}\) in Fig. 3g, which is consistent with the chiral charge +2 of the charge-2 fermions at R. Moreover, the arc-like features are related by a π rotation about \({\tilde{\mathrm{R}}}\) . These properties indicate that the features are surface Fermi arcs. Combining the experimental and calculated results in Fig. 3g, h, the Fermi arcs should extend to the projected FSs around \({\tilde{\mathrm{\Gamma }}}\) of the next surface BZ.Handedness of the surface states in RhSn and CoSi.In Fig. 4, we analyse the handedness of the surface states associated with these chiral fermions. All experimental data in Fig. 4 are analysed in a uniform coordinate system, where kz is specified as the outward normal of the sample surfaces, as illustrated in Fig. 4k. On loop #1, which encloses the (001)-surface projection of the spin-1 fermions in RhSn, bands 1 and 6 pass up through EF from right to the left, while the other band crossings are trivial because of opposite signs of their velocities (Fig. 4a, b). This corresponds to a chiral charge C = −2 of the spin-1 fermions in RhSn in the calculations (Fig. 1g). On loop #2, which encloses the (001)-surface projection of the charge-2 fermions in RhSn, two surface bands pass up through EF from left to right (Fig. 4c, d), corresponding to a chiral charge C = +2 of the charge-2 fermions in RhSn. Furthermore, on loop #3, which encloses the (110)-surface projection of the charge-2 fermions in RhSn, two surface bands pass up through EF from left to right (Fig. 4e, f). Figure 4f also shows some weak features near EF marked as I1 and I2, which are related to the FSs around \({\tilde{\mathrm{M}}}\) adjacent to loop #3. The results on loops #2 and #3 indicate that the surface states on different surfaces associated with the same chiral fermions have the same handedness.Fig. 4Opposite handedness of helical surface states between CoSi and RhSn. a, c, e, g, i ARPES intensity maps at EF around \({\bar{\mathrm{\Gamma }}}\) (a), \({\bar{\mathrm{M}}}\) (c) and \({\tilde{\mathrm{R}}}\) (e) of RhSn, and \({\bar{\mathrm{\Gamma }}}\) (g) and \({\bar{\mathrm{M}}}\) (i) of CoSi. The red dashed ellipses and arrows indicate the momentum paths and directions of loops #1 to #5. b, d, f, h, j ARPES intensity maps and corresponding MDCs showing the chiral surface bands on loops #1 (b), #2 (d), #3 (f), #4 (h) and #5 (j). The bands passing through EF are marked with numbers. The red and black numbers represent nontrivial and trivial bands, respectively. The weak intensities in f, which are related to the FSs around \({\tilde{\mathrm{M}}}\) adjacent to loop #3, are marked as I1 and I2. k Schematics of the relationship between chiral surface bands and chiral charges in the specified coordinate system. The ARPES data in a–h were collected with hν = 75 eV, and those in i, j were collected with hν = 110 eV. Note that the images for CoSi in g–j are mirrored as compared with those in Fig. 4 of ref. 31, where kz was defined as the inward normal of sample surface and is reserved to the current definition.Full size image In the previous study, we have observed helical surface states on the (001) surface of CoSi31. By comparing the handedness of the surface states, we can determine whether the corresponding chiral fermions between CoSi and RhSn have opposite chiral charges. We present the (001)-surface state data of CoSi in Fig. 4g–j. On loop #4, which encloses the (001)-surface projection of the spin-1 fermions in CoSi, bands 1 and 4 pass up through EF from left to right, while the other band crossings are trivial. On loop #5, which encloses the (001)-surface projection of the charge-2 fermions in CoSi, two surface bands pass up through EF from right to left. It is clear that the surface bands have opposite handedness for the corresponding chiral fermions, indicating that the chiral fermions are reversed between RhSn and CoSi. Based on these results, we illustrate in Fig. 4k the relationship between the signs of chiral charges and the handedness of concomitant surface states, where positive and negative chiral charges correspond to right- and left-hand surface states, respectively.

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