Author Correction: Observation of Weyl fermions in a magnetic non-centrosymmetric crystal

The absence of inversion symmetry in non-centrosymmetric materials has a fundamental role in the emergence of a vast number of fascinating phenomena, like ferroelectricity, second harmonic generation, and Weyl fermions. The removal of time-reversal symmetry in such systems further extends the variety of observable magneto-electric and topological effects. Here we report the striking topological properties in the non-centrosymmetric spin-orbit magnet PrAlGe by combining spectroscopy and transport measurements. By photoemission spectroscopy below the Curie temperature, we observe topological Fermi arcs that correspond to projected topological charges of ±1 in the surface Brillouin zone. In the bulk, we observe the linear energy-dispersion of the Weyl fermions. We further observe a large anomalous Hall response in our magneto-transport measurements, which is understood to arise from diverging bulk Berry curvature fields associated with the Weyl band structure. These results establish a novel Weyl semimetal phase in magnetic non-centrosymmetric PrAlGe. The search for magnetic Weyl fermion remains a challenge. Here, the authors report angle-resolved photoemission spectroscopy and magnetotransport measurements resolving the topological properties of Weyl fermion quasiparticles in magnetic non-centrosymmetric crystal PrAlGe.

D evelopment in the search for materials with topological electronic properties has rapidly progressed in the past decade. With a refined understanding of the role symmetries have on the electron wavefunctions Berry curvature, the experimental study of new and exotic quantum phenomena has now become widely accessible [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] . A well-recognized example is the breaking of time-reversal symmetry in magnetic materials, which may result in producing Berry curvature fields that generate an intrinsic anomalous Hall response. Along similar lines, the breaking of inversion symmetry in noncentrosymmetric materials is understood to be essential for fostering new quantum phenomena, such as non-local gyrotropic effects 19 , quantum nonlinear Hall effects 20 , photogalvanic effects 21 , accidental two-fold band degeneracies (Weyl fermions) that are protected by a quantized non-zero integer Chern number (chiral charge) [22][23][24][25] , and anomalous transport 26 .
In this study, we observe that magnetic non-centrosymmetric PrAlGe 14 hosts the emergent topological properties of Weyl fermions by photoemission-based spectroscopy and magnetotransport. In contrast to previous works on magnetic Weyl semimetal candidates Mn 3 Sn 27,28 and Co 3 Sn 2 S 2 29-31 (both centrosymmetric), PrAlGe is calculated to exhibit Weyl fermions in proximity to the Fermi level, making it more suitable for experimentally probing its Berry curvature properties and exploring the connection between photoemission-based band structure and transport. In addition, because PrAlGe lacks both inversion and time-reversal symmetry it can uniquely induce quantum spin currents without a concomitant charge current 14,15 . Motivating future studies on PrAlGe, we experimentally resolve the key topological properties of Weyl fermions by relying predominately on our measurements 11,32 .

Results
Magnetic and electronic properties. PrAlGe crystallizes in a body-centered tetragonal Bravais lattice with space group I4 1 md (No. 109). The basis consists of two Pr, two Al and two Ge atoms, Fig. 1a inset. Along the (001) direction, each atomic layer is comprised of one element, and the layer is shifted relative to the one below by half a lattice constant in either the x or y direction. Single crystal X-ray diffraction suggests that our samples possess the correct lattice structure and lack inversion symmetry (Supplementary Table 1). Measurements of magnetic susceptibility as a function of temperature were fitted to the inverse Curie-Weiss law. The obtained positive Weiss constant indicates the presence of ferromagnetic interactions, Fig. 1a. A direct measurement, to be discussed below, shows that PrAlGe is ferromagnetic with Curie temperature T C = 16 K. The ferromagnetic ground state arises from the spin-polarized f-electron states that are locally coupled and aligned along the c axis, rendering the conduction electron bands near the Fermi level spin-polarized. This is reflected in our ab initio band structure calculations without spinorbit coupling (SOC), in which it also shows that PrAlGe has a semi-metallic profile (top panel: Fig. 1b). The inclusion of SOC interactions couples the spin-up and spin-down states and slightly perturbs the electronic bands (bottom panel: Fig. 1b). The absence of inversion and time-reversal symmetry both contribute to band-splitting at generic crystal momenta. Kramers degeneracy splitting is linked to magnetism in the crystal. Of the Weyl fermions predicted in ferromagnetic PrAlGe 14 , two groups (labeled W 3 and W 4 ) are within ±20 meV of the Fermi level. The ab initio calculated Fermi surface for ferromagnetic PrAlGe predicts the presence of topological Fermi arcs in each quadrant, with an asymmetry across the Γ À M surface high-symmetry line, Fig. 1c VUV-ARPES study of the (001) surface electronic structure. Motivated by our ab initio calculations and transport measurement observing ferromagnetism, we use angle-resolved photoemission spectroscopy (ARPES) measurements at low-photon energies (VUV-ARPES) to map the band structure of PrAlGe on the (001) surface at temperature T = 11 K, below T C . We observe that the ab initio calculation is qualitatively consistent with the measured Fermi surface, Fig. 1e, f. On constant-energy contours of varying binding energy, we observe the following dominant features, Fig. 2a: two concentric closed contours around the Γ point, a distinct "U" shaped state (marked by a guide to the eye, Fig. 2b), and additional surface states near the Y of the surface BZ. The inner closed concentric contour shrinks with deeper binding energy, showing a clear electron-like behavior. To better understand the nature of the "U" state and the spectral intensity in its vicinity, we study an energy-momentum cut through this state, Fig. 2c. We plot the Lorentzian-fitted momentum distribution curves (MDCs) at different binding energies and find that the "U" state disperses towards the Fermi level while a nearby band approaches E F and then turns back toward deeper binding energies. We also find that the "U" state exhibits negligible photon energy dependence, suggesting that it is a surface state (Fig. 2d, Supplementary Fig. 4). A comparison with the ab initio calculated Fermi surface further suggests that the "U" state corresponds to the predicted Fermi arc connecting W 3 and W 4 . The surface state nature of the "U" state and its correspondence with calculation suggests that the ARPES-measured state is a topological Fermi arc.
Topological Fermi arcs and projected chiral charges. To further explore the Fermi arc candidate we search for direct spectroscopic signatures of chiral charges in PrAlGe, taking advantage of the bulk-boundary correspondence between bulk Weyl fermions and surface Fermi arcs 11,32 . We study chiral edge modes along straight and loop energy-momentum cuts (Fig. 3a, b), and present a two-dimensional curvature plot of the measured Fermi surface to further highlight the momentum space trajectory of the Fermi arc candidate, Fig. 3c. A horizontal momentum cut at We observe signatures of a left-moving and right-moving mode related by mirror symmetry. A second derivative plot of the dispersion map further confirms the observed modes and suggests additional neighboring bulk bands which approach E F and then turn back towards deeper binding energies, Fig. 3e. A comparison of this spectrum with the locations of the predicted Weyl fermions suggests that we can interpret the left-and rightmoving modes as two chiral edge modes, associated with Chern number n = ±1, Fig. 3b. In this way, the momentum cut is associated with a 2D momentum-space slice carrying Chern number n tot = 0, since n l = −1 and n r = +1. This again suggests that the left-and right-moving modes giving rise to two mirror partnered "U" states are topological Fermi arcs.
To provide further evidence of chiral charges in PrAlGe, we next perform an analysis of edge modes along closed loops in the surface BZ (black dashed circles labelled P and M, Fig. 3c). By counting chiral edge modes on these circular paths, we search for evidence of n tot ≠ 0. Unrolling the energy-momentum dispersion for loop P, we observe one left-moving chiral mode that is dispersing towards E F , Fig. 3f. This result unambiguously shows chiral charge −1 on the associated bulk manifold. Analogously, along loop M we observe a right-moving chiral mode dispersing towards E F , Fig. 3g. By Lorentzian fitting of the MDCs along loop M at varying binding energies, we again observe a right-moving mode, Fig. 3h, suggesting that the corresponding bulk manifold encloses chiral charge +1. As an additional check, the ab initio band dispersion calculation along k y = −0.25 (2π/a) shows a right-moving chiral mode dispersing toward E F , Fig. 3i. An overlay of the Lorentzian fits of the chiral mode on the calculated band dispersion shows a match between our results (Supplementary Fig. 5). In this way, our low-energy ARPES spectra directly resolve topological Fermi arcs and demonstrate chiral charges in PrAlGe through the bulk-boundary correspondence 11,32 . Further, the observed Fermi arc asymmetry across the Γ À M is consistent with our ab initio calculations for PrAlGe (Supplementary Fig. 2).
Bulk Weyl cone dispersion. Next, we provide a comparison between the experimental bulk band structure obtained by ARPES at soft X-ray energies (SX-ARPES) and our ab initio calculations (Fig. 4). To demonstrate the linear dispersion of the bulk Weyl cones, our analysis looks at the energy-dispersion maps along various horizontal and vertical paths that intersect the ab initio calculated positions of the Weyl fermions on the k z = 0 plane, Fig. 4a. The large probing depth provided by SX-ARPES allows for a comparison between the ab initio calculated (Fig. 4b) and experimentally measured (Fig. 4c) bulk Fermi surface on the k z = 0 plane. Both are qualitatively consistent and show an absence of asymmetry across the Γ À M line. Along vertical cut 1 (green line: Fig. 4d) and horizontal cut 2 (blue line: Fig. 4e), we observe the linear dispersion of the W 3 and W 4 Weyl cones. Horizontal cut 3 (red line: Fig. 4f) further confirmes the linear dispersion of the W 4 Weyl fermion. Second derivative plots of Cuts 1-3 with guides to the eye for the Weyl cones further illustrate their linear dispersion (Fig. 4g-i). Within the resolution of our measurements, the W 3 and W 4 Weyl fermions are located on the k z = 0 plane at (0.15, −0.32) 2π/a and (0.13, −0.22) 2π/a, respectively. Due to our SX-ARPES measurement temperature being comparable to the Curie temperature of our PrAlGe samples, and limited SX-ARPES energy resolution or spectral linewidth, our data did not clearly resolve the Zeeman splitting, suggesting that further experimental work is needed. Howbeit, the qualitative agreement between our SX-ARPES measurements (Fig. 4d-i) and ab initio calculations (Fig. 4j-l) suggests the observation of bulk Weyl cones in PrAlGe.
Anomalous Hall transport. Having experimentally demonstrated topological Fermi arcs and bulk Weyl cones in PrAlGe, we next investigate additional phenomena mediated by Berry curvature using magneto-transport ( Supplementary Fig. 6). We study the magnetization M as a function of magnetic induction μ 0 H (Fig. 5a) and observe that PrAlGe is a soft ferromagnet with an easy axis along the c-direction. Furthermore, we find that the transverse resistivity ρ yx exhibits an anomalous Hall effect, described by ρ yx = R H B + μ 0 R S M, where R H is the ordinary (Lorentz-force) Hall coefficient and R S is the anomalous Hall coefficient 33 . As shown in the inset of Fig. 5b, R S (zero for high T) grows rapidly toward large values while the small value for R H (almost invariant for different T) decreases very quickly below T C . The observed behavior for R H and R S suggests that the anomalous Hall effect arises near T C . The measured R S coefficient reaches a saturation value of about 1.5 μΩ cm T −1 at 2 K, where it dominates the response 34,35 . To investigate the origin of the observed behavior for the anomalous Hall effect, we plotted the anomalous Hall contribution ρ A yx as a function of carrier concentration p = 1/ eR H , where e is the charge of an electron. The result shows clustered values in the vicinity of 1 μΩcm for different samples, see Fig. 5c inset. To better understand the origin of ρ A yx , we calculated the Berry curvature contribution to the anomalous Hall conductivity, the so-called intrinsic anomalous Hall conductivity, σ A int yx as a function of carrier concentration, Fig. 5c. The calculation predicts a roughly carrier concentration-independent value of~600 Ω −1 cm −1 with carrier densities from p = 0.9 to 1.7 × 10 21 cm −3 . This corresponds to an intrinsic contribution to the anomalous Hall resistivity of ρ yx ρ 2 0 % 0:6 μΩ cm, which we plot as a horizontal green line in inset, Fig. 5c. We find a remarkable agreement with the measured ρ A yx % 1μΩ cm. Bulk-boundary-transport correspondence in PrAlGe. To study the origin of the Berry curvature fields giving rise to the intrinsic anomalous Hall response, we compare the ARPES-measured Fermi surface with the calculated Berry curvature field. By summing over energies below E F , the Berry curvature magnitude |Ω (k)| at k z = 0 shows concentrated hot spots, see Fig. 5d. The points of concentrated Berry curvature correspond to the position of the W 3 and W 4 Weyl fermions. A qualitative comparison with the ARPES-measured Fermi surface shows that within our momentum-space resolution the hot spot region coincides with the termination points of the measured topological Fermi arc. According to ref. 36 , the AHE of an ideal magnetic Weyl semimetal with only one pair of Weyl fermions near the Fermi level can be written as: σ Weyl = e 2 k/2πh, where k is their momentum space separation. A quantitative estimate using the measured Weyl fermion separation k Weyl ≈ 0.15 Å −1 yields an intrinsic anomalous hall conductivity σ Weyl ≈ 738 Ω −1 cm −1 , consistent with our ab initio calculations. Collectively, our results provide strong evidence suggesting that the measured intrinsic anomalous Hall response arises from the Berry curvature contributions of the Weyl fermions in magnetic non-centrosymmetric PrAlGe 36 .

Discussion
The captured surface-and bulk-states in our ARPES spectra and magneto-transport measurements, taken together with support from ab initio calculations, establish a surface-bulk-transport correspondence demonstrating that PrAlGe exhibits novel Berry curvature mediated topological electronic phenomena. Our results open new research directions in understanding and engineering tunable topological electronic properties in the noncentrosymmetric magnet PrAlGe. In particular, due to the absence of both time-reversal and inversion symmetry, exotic types of photogalvanic effects may emerge [37][38][39][40] . Additionally, topological currents in PrAlGe may also allow for the development of all-electrical spin generation and injection with no entropy production. Lastly, the soft-ferromagnetism may allow the spin-polarized topological currents to be turned on/off via an external magnetic field 41  non-centrosymmetric PrAlGe an exciting material platform for probing topological and quantum matter physics.

Methods
Single crystal growth. Single crystals of PrAlGe were grown by the self-flux technique with purified Pr ingots (99.9%), Al shots (99.99%), and Ge pieces (99.99%). Stoichiometric mixtures of Pr1Al18Ge1 were set in alumina crucibles and then sealed in fused silica ampoules under partially filled argon atmosphere. After being pegged at 1150°C for a few hours, the ampoules were slowly cooled down to the centrifugal temperature 750°C at a rate of 0. 1°C/min.
Angle-resolved photoemission spectroscopy. Low-energy ARPES measurements (VUV-ARPES) were carried out at Beamlines (BL) 5-2 of the Stanford Synchotron Radiation Lightsource (SSRL) at SLAC in Menlo Park, CA, USA, with a Scienta R4000 electron analyzer. The angular resolutions was better than 0.2°, and the energy resolution was better than 20 meV. The beam spot size was about 20 × 40 μm 2 . Samples were cleaved in situ and measured under vacuum better than 5 × 10 −11 Torr and temperatures <11 K. Soft X-ray ARPES measurements (SX-ARPES) were performed at the ADRESS beamline at the Swiss Light Source in the Paul Scherrer Institut (PSI) in Villigen, Switzerland. The combined (beamline and analyzer) energy resolution of the SX-ARPES measurements varied between 40 and 80 meV. The angular resolution of the SX-ARPES analyzer was better than 0.2°. PrAlGe samples were cleaved in situ under a vacuum condition better than 5 × 10 −11 Torr and temperature less than the Curie temperature T C .
Magnetization measurements. Magnetization measurements were performed using the Quantum Design Magnetic Property Measurement System (MPMS-3).
Transport measurements. Resistance and Hall effect measurements were performed in a Quantum Design Physical Property Measurement System (PPMS), using the standard four-probe technique with silver paste contacts that were cured at room temperature.
First-principles calculations. The first-principles calculations were performed within the density functional theory (DFT) framework using the projector augmented wave method as implemented in the VASP package 43,44 . The generalized gradient approximation (GGA) was used 45 for the exchange-correlation effect and the Hubbard energy U used in the calculation is 4 eV. A Γ-centered k-point 14 × 14 × 14 mesh was used and spin-orbit coupling (SOC) was included in selfconsistent cycles. To generate the (001)-surface states of PrAlGe, Wannier functions were generated using the d and f orbitals of Pr, and the s and p orbitals of Al and Ge. The surface states were calculated for a semi-infinite slab by the iterative Green's function method. They were optimized based on experimental results 46 .

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.