Network of Topological Nodal Planes, Multifold Degeneracies, and Weyl Points in CoSi.

We showcase the importance of global band topology in a study of the Weyl semimetal CoSi as a representative of chiral space group (SG) 198. We identify a network of band crossings comprising topological nodal planes, multifold degeneracies, and Weyl points consistent with the fermion doubling theorem. To confirm these findings, we combined the general analysis of the band topology of SG 198 with Shubnikov-de Haas oscillations and material-specific calculations of the electronic structure and Berry curvature. The observation of two nearly dispersionless Shubnikov-de Haas frequency branches provides unambiguous evidence of four Fermi surface sheets at the R point that reflect the symmetry-enforced orthogonality of the underlying wave functions at the intersections with the nodal planes. Hence, irrespective of the spin-orbit coupling strength, SG 198 features always six- and fourfold degenerate crossings at R and Γ that are intimately connected to the topological charges distributed across the network.

Symmetry-enforced topological nodal planes at the Fermi surface of a chiral magnet.

Despite recent efforts to advance spintronics devices and quantum information technology using materials with non-trivial topological properties, three key challenges are still unresolved1-9. First, the identification of topological band degeneracies that are generically rather than accidentally located at the Fermi level. Second, the ability to easily control such topological degeneracies. And third, the identification of generic topological degeneracies in large, multisheeted Fermi surfaces. By combining de Haas-van Alphen spectroscopy with density functional theory and band-topology calculations, here we show that the non-symmorphic symmetries10-17 in chiral, ferromagnetic manganese silicide (MnSi) generate nodal planes (NPs)11,12, which enforce topological protectorates (TPs) with substantial Berry curvatures at the intersection of the NPs with the Fermi surface (FS) regardless of the complexity of the FS. We predict that these TPs will be accompanied by sizeable Fermi arcs subject to the direction of the magnetization. Deriving the symmetry conditions underlying topological NPs, we show that the 1,651 magnetic space groups comprise 7 grey groups and 26 black-and-white groups with topological NPs, including the space group of ferromagnetic MnSi. Thus, the identification of symmetry-enforced TPs, which can be controlled with a magnetic field, on the FS of MnSi suggests the existence of similar properties-amenable for technological exploitation-in a large number of materials.