Fig. 4: First-principles calculations in PrAlGe.

a, b Schemes of BZ indicating the locations of the Weyl nodes in the PM and FM state¹⁰. c–g First-principles calculation of the electronic band structure with SOC in the PM (c, d) as well as FM (e–g) state along k-paths in BZ, near some specific Weyl nodes (i.e., black stars PM1 and PM2 as well as FM1, FM2 and FM3 in a, b and i for the PM and FM state, respectively). The coloured bars in c–g point out a selection of representative interband transitions, then mapped onto Δσ₁(ω). The green and red bars refer to the transitions in the PM state, while the blue ones to those in the FM state. h Calculation of the interband part of σ₁(ω) (i.e., Δσ₁(ω)) in both the PM and FM state, derived from the electronic band structure (Supplementary Equation 3). The colour code for the arrows matches the one of the bars in c, d, thus establishing their relationship. i Effective Berry curvature (i.e., its difference between the PM and FM state, Supplementary Fig. 9) integrated along k_z as a function of k_x and k_y from our ab initio calculations, which emphasises its strong enhancement in the FM state at those k-points of BZ almost coincident with the (projected) Weyl nodes (green symbols). Black stars foreground the selected Weyl nodes in the FM state, as in b. We refer to Supplementary Notes 6, 7 and 8 for details about all calculations.