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 momentum k. 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.

相对论量子场论中，Weyl费米子最初被提出为无质量的狄拉克费米子。在固体中，Weyl费米子常在时间反转对称或空间反转对称被破坏时出现，这会在每个晶体动量k处提升布洛赫状态的克莱默简并性。固体中的维尔费米子具有两个新颖的特征：自旋动量锁定和发散的贝里曲率，对应于动量空间中形成的磁单极子。这两种效应带来了许多新颖的传输现象，即磁激发和非线性光学效应。在这篇综述中，我们讨论了各种系统中的一些现象，特别是3D金属铁磁体，过渡金属氧化物半导体和半金属中的异常霍尔效应和磁振子激发。拓扑绝缘子的二维磁面上的量子异常霍尔效应，轴绝缘子态和磁天文离子；非中心对称Weyl材料中的非线性现象。