Transport, magnetic and optical properties of Weyl materials

Weyl fermions were originally proposed as massless Dirac fermions in relativistic quantum field theory. In solids, Weyl fermions often appear when either time-reversal symmetry or spatial-inversion symmetry is broken, which lifts the Kramer's degeneracy of Bloch states at each crystal momentumk. Weyl fermions in solids are characterized by two novel features: spin-momentum locking and a diverging Berry curvature corresponding to a magnetic monopole forming in momentum space. These two effects bring about many novel transport phenomena, magnetic excitations and nonlinear optical effects. In this Review, we discuss some of these phenomena in various systems, in particular, the anomalous Hall effect and magnon excitation in 3D metallic ferromagnets, transition-metal-oxide semiconductors and semimetals; the quantum anomalous Hall effect, axion insulator state and magnetic skyrmions at the 2D magnetic surface of topological insulators; and nonlinear phenomena in noncentrosymmetric Weyl materials. Weyl fermions in solids exhibit unique features such as spin-momentum locking and a diverging Berry curvature corresponding to a magnetic monopole in momentum space. This Review surveys the transport phenomena, magnetic excitations and nonlinear optical effects observed in these systems.