Topological band crossings in energy spectra are connected to the emergence of prominent effects, for example, the anomalous Hall effect and Fermi arc surface states, which appear as open Fermi surfaces on the boundary of the system. In this talk I will present symmetry-based approaches to infer the existence of various point, line, and plane crossings as well as weak insulator topology.
Crystalline symmetries do not just protect but may also enforce a specific band topology. I will discuss the close connection between the chirality of point crossings and the rotation eigenvalues. While the existence of crossings pinned to high-symmetry positions can be easily obtained from the representations of the corresponding little groups, I will give some examples of movable crossings within the Brillouin zone. These results can then be combined to infer, for example, the topology of nodal planes.
Further, I will present some of our efforts to give a comprehensive overview of enforced band structures in orthorhombic and tetragonal space groups, including, for example, almost movable nodal lines, nodal chains, and (topological) nodal planes. We propose example materials and I will touch upon the topological nodal planes in ferromagnetic MnSi, and double Weyl points in NbO2/TaO2.

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