Topology is a global aspect of materials, leading to fundamental new properties for compounds with large relativistic effects. The incorporation of heavy elements gives rise to non-trivial topological phases of matter, such as topological insulators, Dirac and Weyl semimetals. The semimetals are characterized by band-touching points with linear dispersion, similar to massless relativistic particles in high energy physics.
The interplay of symmetry, relativistic effects and, in magnetic materials, the magnetic structure, allows for the realization of a wide variety of topological phases through Berry curvature design. The Berry curvature describes the entanglement of the valence and conduction bands in an energy band structure. Weyl points and other topological electronic bands can be manipulated by various external perturbations like magnetic fields and pressure, which results in exotic local properties such as the chiral or gravitational anomaly and large topological Hall effects, concepts which were developed in other fields of physics such as high energy physics and astrophysics.