Fig. 1. Topology and BZ symmetry of LaAlGe.

(A) Body-centered tetragonal structure of LaAlGe, with space group I4₁md (109). The structure consists of stacks of La, Al, and Ge layers, and along the (001) direction, each layer consists of only one type of element. (B) The bulk and (001) surface BZ. (C) First-principles band structure calculations along high-symmetry directions without spin-orbit coupling (SOC). (D) Momentum space configuration of the four nodal lines (two on the k_x = 0 and two on the k_y = 0 mirror planes) denoted by the rings, as well as the four spinless pairs of Weyl nodes denoted as W₃ on the k_z = 0 plane, in the absence of SOC. Blue and red colors indicate positive and negative chiralities, respectively. (E) Configuration of the 40 Weyl nodes in the bulk BZ created upon the inclusion of SOC. The nodal lines are gapped out by SOC, and 24 Weyl nodes emerge in the vicinity of the nodal lines. In addition, each spinless W₃ Weyl node splits into two spinful Weyl nodes of the same chirality, which we denote as W₃′ and W₃″. Hence, the eight W₃ without SOC evolve into eight W₃′ and eight W₃″ Weyl nodes with SOC. Therefore, in total, there are 40 Weyl nodes. For the 24 Weyl nodes that emerge from the gapping of the nodal line, we denote the 8 Weyl nodes that are near the boundaries of k_z = 0 plane as W₁ and the other 16 that are away from the k_z = 0 plane as W₂. The W₃′ and W₃″ are also on the k_z = 0 plane, but they are near the diagonal lines. (F) Projection of the Weyl nodes on the (001) surface BZ in one quadrant. (G) Schematics comparing the three types of Weyl nodes appearing upon the inclusion of SOC. The W₂ nodes are type II Weyl nodes, whereas the W₁, W₃′, and W₃″ nodes are type I. W₂ Weyl nodes are located almost exactly at the Fermi level, whereas W₁, W₃′, and W₃″ Weyl nodes are about 60, 110, and 130 meV above the Fermi level, respectively. (H) Core level measurement of the studied samples, which clearly shows the expected La, Al, and Ge peaks. a.u., arbitrary units.