The search for materials exhibiting topological electronic properties has significantly advanced in the last decade. A deeper understanding of the influence of symmetries on electron wavefunctions and Berry curvature has made the experimental investigation of exotic quantum phenomena more accessible. Breaking time-reversal symmetry in magnetic materials can generate Berry curvature fields leading to an intrinsic anomalous Hall response. Similarly, breaking inversion symmetry in non-centrosymmetric materials is crucial for phenomena like non-local gyrotropic effects, quantum nonlinear Hall effects, photogalvanic effects, and Weyl fermions protected by a quantized Chern number. This study focuses on PrAlGe, a magnetic non-centrosymmetric material, to experimentally investigate the presence of Weyl fermions and their associated topological properties. Unlike previous studies on magnetic Weyl semimetal candidates (Mn3Sn and Co3Sn2S2, both centrosymmetric), PrAlGe is predicted to have Weyl fermions close to the Fermi level, making it ideal for probing Berry curvature and the link between photoemission-based band structure and transport. The absence of both inversion and time-reversal symmetry in PrAlGe uniquely enables quantum spin currents without a corresponding charge current, motivating further research.

The literature extensively covers the theoretical predictions and experimental observations of Weyl fermions and related topological phenomena in various materials. Studies on magnetic Weyl semimetals, such as Mn3Sn and Co3Sn2S2, have provided valuable insights into the anomalous Hall effect and other transport properties arising from Berry curvature. However, the proximity of Weyl points to the Fermi level and the interplay between broken time-reversal and inversion symmetries are crucial factors influencing experimental observability. Theoretical work on PrAlGe predicted the presence of Weyl fermions near the Fermi level, making it a promising candidate for experimental verification. The literature also discusses the connection between topological Fermi arcs (surface states) and bulk Weyl fermions, establishing a bulk-boundary correspondence essential for understanding the topological nature of these materials. Prior work has emphasized the importance of angle-resolved photoemission spectroscopy (ARPES) in identifying topological Fermi arcs and the anomalous Hall effect as a signature of Berry curvature in transport measurements.

This research employed a multi-pronged approach combining experimental techniques and theoretical calculations. Single crystals of PrAlGe were grown using the self-flux technique. Angle-resolved photoemission spectroscopy (ARPES) was used to map the electronic band structure. Low-energy ARPES (VUV-ARPES) with low photon energies was employed to study the surface electronic structure at low temperatures (below the Curie temperature Tc), allowing for the identification of topological Fermi arcs. Soft X-ray ARPES (SX-ARPES) with higher photon energies provided greater probing depth, enabling the observation of the bulk band structure and Weyl cones. The magnetic properties were characterized by magnetization measurements using a Quantum Design Magnetic Property Measurement System (MPMS-3), determining the Curie temperature. Magneto-transport measurements using a Quantum Design Physical Property Measurement System (PPMS) with a standard four-probe technique explored the anomalous Hall effect. First-principles calculations within the density functional theory (DFT) framework using the projector augmented wave method (as implemented in the VASP package) were performed to predict the band structure, Fermi surface, and Berry curvature. The generalized gradient approximation (GGA) was used for exchange-correlation effects, incorporating a Hubbard U parameter of 4 eV and spin-orbit coupling (SOC) in self-consistent calculations. Wannier functions were generated to calculate the (001) surface states for a semi-infinite slab using the iterative Green's function method, optimized based on experimental findings. Chiral edge modes were analyzed along straight and loop energy-momentum cuts in ARPES data to determine the topological charge.

The key findings of this study demonstrate the existence of Weyl fermions in PrAlGe through a combination of experimental observations and theoretical calculations. VUV-ARPES measurements revealed the presence of topological Fermi arcs, exhibiting a characteristic "U" shape. The negligible photon energy dependence of these arcs confirmed their surface state nature. Further analysis of chiral edge modes along various energy-momentum cuts revealed enclosed Chern numbers of ±1, providing direct spectroscopic evidence of chiral charges. SX-ARPES measurements, benefiting from their greater probing depth, revealed the linear energy dispersion of bulk Weyl cones, consistent with theoretical predictions. Magneto-transport measurements showed a large anomalous Hall effect, whose magnitude and behavior strongly suggested its origin in the intrinsic Berry curvature associated with the Weyl fermions. The calculated intrinsic anomalous Hall conductivity showed remarkable agreement with the experimental value, supporting the dominance of the Berry curvature contribution. A surface-bulk-transport correspondence was established by comparing the ARPES-measured Fermi surface with the calculated Berry curvature, revealing concentrated Berry curvature fields near the Weyl fermion projections and the termination points of the Fermi arcs. The quantitative estimate of the intrinsic anomalous Hall conductivity using the measured Weyl fermion separation was consistent with both ab initio calculations and experimental results.

The findings directly address the hypothesis of the existence of Weyl fermions in PrAlGe. The combined experimental evidence from ARPES (both VUV and SX) and magneto-transport measurements, corroborated by theoretical calculations, strongly supports the presence of Weyl fermions and their associated topological properties. The large anomalous Hall effect observed is a direct consequence of the significant Berry curvature associated with the Weyl points. The excellent agreement between the experimental and calculated anomalous Hall conductivity highlights the dominance of the intrinsic Berry curvature contribution. This work extends our understanding of topological materials beyond centrosymmetric systems, showcasing the importance of both broken inversion and time-reversal symmetries in realizing Weyl semimetal phases. The observed surface-bulk-transport correspondence solidifies the understanding of the connection between surface Fermi arcs and bulk Weyl fermions, supporting the topological nature of the system.

This study successfully demonstrated the existence of Weyl fermions and related topological phenomena in the non-centrosymmetric magnetic material PrAlGe. The combined experimental and theoretical findings establish a novel Weyl semimetal phase in this material. Future research directions include investigating exotic photogalvanic effects arising from the lack of both time-reversal and inversion symmetry and exploring potential applications of spin-polarized topological currents for all-electrical spin generation and injection with minimal entropy production. The soft ferromagnetism of PrAlGe offers the possibility of external magnetic field control of these topological currents.

The limited energy resolution in SX-ARPES measurements, especially near the Curie temperature, prevented the clear resolution of Zeeman splitting of the Weyl cones. Further experiments with improved resolution might provide a more precise determination of the Weyl fermion properties. The analysis of the anomalous Hall effect primarily focused on the intrinsic contribution, neglecting other possible extrinsic contributions. A more thorough investigation separating intrinsic and extrinsic contributions could provide a more complete understanding of the Hall effect in PrAlGe.