When Paul Dirac introduced his famous equation for relativistic fermions in 1928, he aimed to describe one well-known particle: the electron. Shortly thereafter, Hermann Weyl observed that the equation has a special solution when the mass is set to zero. The so-called Weyl fermions embodied by that solution would be charged, like electrons, but being massless, they would travel faster and with less energy dissipation. The particles would also be chiral, like neutrinos, with each one’s handedness depending on whether its spin is aligned or antialigned with its momentum. Those features make Weyl fermions appealing candidates for use in electronic and spintronic devices.

No such elementary particle has yet been found. However, in 2015 three groups of researchers identified the first Weyl semimetal (WSM), tantalum arsenide, which hosts quasiparticles—collective excitations of electrons—with the properties of Weyl fermions.A WSM must have a broken symmetry, and in TaAs, it’s inversion symmetry. Researchers, however, have continued searching for materials, particularly ferromagnetic materials, that instead rely on broken time-reversal symmetry. Tying a WSM crystal’s properties to magnetism, which can be adjusted using temperature changes or external fields, makes them potentially tunable.

Three new papers provide experimental evidence for magnetic WSMs. Yulin Chen’s team at Oxford University and Haim Beidenkopf’s team at the Weizmann Institute of Science, together with collaborators,presented studies of CoSn, and Zahid Hasan’s group at Princeton Universitylooked at CoMnGa. The works identify important features in the electronic structures of both materials’ bulk and surface states.

Electronic underpinnings Section: Choose Top of page ABSTRACT Electronic underpinnings << Hunting for quasiparticle... References CITING ARTICLES

4 et al. , Annu. Rev. Condens. Matter Phys. 8, 289 (2017); Annu. Rev. Condens. Matter Phys. 8, 337 (2017). 4. M. Z. Hasan, 289 (2017); https://doi.org/10.1146/annurev-conmatphys-031016-025225 B. Yan, C. Felser,, 337 (2017). https://doi.org/10.1146/annurev-conmatphys-031016-025458 1 Physics Today, The secret to a WSM’s behavior is in its band structure, which has similar origins to that of a topological insulator.In both cases, interactions cause the conduction and valence bands to invert near the Fermi surface. Spin–orbit coupling then opens a gap between them, as illustrated in figure. In topological insulators, it opens a bandgap throughout the bulk. (See April 2009, page 12 .) But in a WSM, the valence and conduction bands still touch at a set of points.

A WSM’s band structure is similar to a three-dimensional version of graphene. In both materials the dispersion relation is linear around the bands’ contact points, so low-energy electron excitations travel at a constant speed set by the dispersion relation’s slope. Having a constant speed that doesn’t depend on energy makes the excitations effectively massless. But that doesn’t mean they travel at the speed of light—they are still about two orders of magnitude slower than photons.

Physics Today, In graphene, the points at which the valence and conductance bands meet are degenerate because the system is invariant under both inversion and time-reversal symmetry (see the article by Andre Geim and Allan MacDonald, August 2007, page 35 ). Known as Dirac points, they describe excitations of both chiralities, so the momentum and spin can be either parallel or antiparallel. But in a WSM, the presence of a broken symmetry lifts that degeneracy and splits the Dirac points into pairs of Weyl nodes with opposite chirality.

A WSM that breaks inversion symmetry but preserves time-reversal symmetry must have at least four Weyl nodes because time reversal flips the signs for both momentum k and spin s. If (k,s) describes a Weyl node, then under time reversal, so does (−k,−s). But those have the same chirality, so (k,−s) and (−k,s) with opposite chirality must also be included to maintain zero net chirality. If, instead, inversion symmetry is preserved and time-reversal symmetry is broken, there can be just two Weyl nodes. Inversion symmetry changes only the sign of the momentum, so the inversion-symmetric points (k,s) and (−k,s) already have opposite chirality. Therefore, to produce the simplest WSM with only two Weyl nodes—which would be ideal for studying the underlying physics—the material must break time-reversal symmetry.

Physics Today, Weyl nodes in crystal momentum space behave like magnetic monopoles in real space. If an electron made a closed loop around a magnetic monopole, its wavefunction would acquire a nonzero phase. A Weyl node does the same thing. Like a vector sliding along the surface of a mobius strip, the wavefunction’s failure to regain its initial state reflects the nontrivial curvature and topology of the underlying space. (See the article by Joseph Avron, Daniel Osadchy, and Ruedi Seiler, August 2003, page 38 .) Weyl nodes serve as sources and sinks of so-called Berry curvature, and they are associated with nonzero values of a topological invariant known as the Chern number. The topological nature of the Weyl points makes them appealing for electronic applications because it protects the surface states. Perturbations don’t change the underlying topology, so the states aren’t destroyed by moderate deformations or impurities.

Another hallmark of a WSM is the appearance of spin-polarized surface states. In momentum space, the states appear as lines, known as surface Fermi arcs (SFAs), that connect surface projections of pairs of Weyl points with opposite chirality. The SFAs are confined to the surface of the material by the topology of the band structure.