Correspondence between Euler charges and nodal-line topology in Euler semimetals

Real multi-bandgap systems have non-abelian topological charges, with Euler semimetals being a prominent example characterized by real triple degeneracies (RTDs) in momentum space. These RTDs serve as "Weyl points" for real topological phases. Despite theoretical interest, experimental observations of RTDs have been lacking, and studies mainly focus on individual RTDs. Here, we experimentally demonstrate physical systems with multiple RTDs in crystals, analyzing the distribution of Euler charges and their global connectivity. Through Euler curvature fields, we reveal that type I RTDs have quantized point Euler charges, while type II RTDs show continuously distributed Euler charges along nodal lines. We discover a correspondence between the Euler number of RTDs and the abelian/non-abelian topological charges of nodal lines, extending the Poincaré-Hopf index theorem to Bloch fiber bundles and ensuring nodal line connectivity. In addition, we propose a "no-go" theorem for RTD systems, mandating the balance of positive and negative Euler charges within the Brillouin zone.