Weyl FermionEdit
Weyl fermions are among the most intriguing massless fermions in modern physics. They are defined as chiral, spin-1/2 particles that satisfy the Weyl equation, a two-component form of the Dirac equation obtained in the limit where the mass term vanishes. Named after Hermann Weyl, these particles occupy a special place in both high-energy physics and condensed matter, where they can appear as emergent quasiparticles in certain materials. In the standard model of particle physics, the left-handed components of neutrinos were historically treated as Weyl spinors, though the discovery of neutrino mass requires a more complete description that can mix chiral states. Nevertheless, the Weyl concept remains a powerful organizing principle for understanding chirality, symmetry, and topology in quantum systems.
A central development in the last decade has been the realization that Weyl fermions can exist as excitations in solid-state systems, where they are not fundamental particles but quasiparticles that arise from the band structure of crystalline materials. In these systems, the electronic bands cross at points in momentum space called Weyl nodes, with each node carrying a definite chirality. The presence of Weyl nodes in pairs of opposite chirality is a general consequence of the underlying lattice regularization—an example of the Nielsen–Ninomiya no-go theorem. Such materials, known as Weyl semimetals, have opened a bridge between high-energy concepts and experimental condensed matter physics, allowing researchers to observe phenomena originally proposed for relativistic fermions in a tabletop setting. Notable experimental realizations include TaAs, NbAs, TaP, and NbP, where the Weyl nodes and associated surface states have been identified through angle-resolved photoemission spectroscopy and transport measurements. TaAs NbAs TaP NbP ARPES Weyl semimetal
Theoretical foundations
Weyl fermions emerge from the massless limit of the Dirac equation. The Dirac equation describes a relativistic spin-1/2 particle with both left- and right-handed components, but when the mass term is set to zero, the equations decouple into independent left- and right-handed Weyl equations. This decoupling makes chirality a good quantum number in the massless limit, so a Weyl fermion can be described by a two-component spinor rather than a four-component Dirac spinor. The concepts of chirality and helicity become central in this framework, with left-handed and right-handed states transforming differently under Lorentz transformations. For readers exploring the mathematics, the Weyl equation provides a clean setting to study how symmetry, topology, and quantum anomalies arise in relativistic fermion systems. See Dirac equation Weyl equation chirality helicity
Weyl fermions are not isolated normal modes in a lattice unless certain constraints are satisfied. The Nielsen–Ninomiya theorem shows that lattice regularization of a chiral theory cannot produce a single, isolated Weyl node; Weyl nodes must appear in pairs of opposite chirality, ensuring overall topological neutrality. The nodes act as monopoles of Berry curvature in momentum space, carrying a topological charge equal to their chirality. This topology underpins the existence of robust surface states, such as Fermi arcs, which connect projections of Weyl nodes of opposite chirality on the surface Brillouin zone. See Nielsen–Ninomiya theorem Berry curvature Fermi arc
Weyl fermions contrast with Dirac fermions and Majorana fermions. A Dirac fermion combines left- and right-handed components into a single four-component field, while a Majorana fermion is its own antiparticle. The Weyl description emphasizes chirality and the massless limit, which has deep implications for anomaly physics, including the chiral anomaly that can appear when Weyl fermions are subjected to external fields. See Dirac fermion Majorana fermion chiral anomaly
Emergence in condensed matter
In solid-state systems, Weyl fermions are not fundamental particles but low-energy excitations tied to the electronic band structure. A Weyl semimetal features bands that cross at discrete points (Weyl nodes) with linear dispersion in all directions around each node. The separation of nodes in momentum space and their assigned chiralities give rise to a range of unusual transport and optical responses. These materials are topological in nature, with surface states that form open contours—Fermi arcs—that terminate at the projections of Weyl nodes. See Weyl semimetal Weyl node Berry curvature Fermi arc
The discovery of Weyl semimetals was propelled by the synthesis and characterization of materials such as TaAs, NbAs, TaP, and NbP. Experimental probes like ARPES have directly visualized the bulk Weyl nodes and the associated surface Fermi arcs, while transport experiments have explored how Weyl physics manifests in conductivity, magnetoresistance, and optical responses. See TaAs NbAs TaP NbP ARPES
In the lattice context, the Weyl nodes act as sources and sinks of Berry curvature, giving rise to a topological charge that protects the nodes against small perturbations that do not couple nodes of opposite chirality. This robustness underpins the appeal of Weyl semimetals as platforms to study relativistic-like physics in a controlled setting. See Berry curvature Nielsen–Ninomiya theorem
Experimental observations and signatures
Angle-resolved photoemission spectroscopy (ARPES) has been instrumental in imaging the band structure of Weyl semimetals and in locating Weyl nodes and Fermi arcs on the surface. Magnetotransport measurements, including observations of negative longitudinal magnetoresistance in some Weyl materials, have been interpreted as signatures of the chiral anomaly—a hallmark of Weyl physics under parallel electric and magnetic fields. Yet, the interpretation of transport data in solids can be subtle, and alternative explanations such as current jetting or inhomogeneities have been discussed in the literature. See ARPES Fermi arc chiral anomaly magnetoresistance
These experiments also explore how external perturbations—pressure, strain, and chemical substitution—move Weyl nodes in momentum space, potentially tuning the separation of nodes and modifying surface states. The ability to manipulate Weyl nodes in real materials is a key area of research, with implications for both fundamental physics and potential applications in electronics and spintronics. See Weyl semimetal TaAs NbAs
Controversies and debates
As with any frontier in physics, there are debates about interpretation and scope. In condensed matter, some experimental signatures attributed to the chiral anomaly have been challenged by critiques emphasizing that alternative, nonrelativistic mechanisms might mimic similar transport behavior under certain measurement geometries. Ongoing work seeks to disentangle intrinsic Weyl physics from extrinsic effects such as sample inhomogeneity and contact geometry. See chiral anomaly magnetoresistance
On the fundamental side, Weyl fermions as elementary particles remain a theoretical description within the standard model under massless conditions; the observed masses of known fermions imply that exact Weyl behavior cannot persist for all participants at low energies. Neutrinos, for example, are now understood to have mass, and their full description requires mixing between chiral states, which moves beyond the strict Weyl picture. This has not diminished the value of Weyl concepts as organizing principles in both high-energy and condensed matter physics, but it does highlight the distinction between fundamental particles and emergent quasiparticles in solids. See neutrino Dirac equation Weyl equation Majorana fermion
Significance and outlook
Weyl fermions provide a rare link between relativistic quantum field theory and solid-state physics, enabling experimental access to phenomena that were once thought exclusive to high-energy regimes. The study of Weyl semimetals has enriched understanding of topology in materials science, clarified how symmetry and topology constrain electronic structure, and suggested new routes to control electronic properties via external fields and material design. See Weyl semimetal Topological matter Berry curvature