Topological SemimetalEdit
Topological semimetals are quantum materials in which the electronic excitations at low energy mimic relativistic particles and exhibit robust band touchings protected by symmetries. Unlike conventional metals or insulators, these systems host massless or nearly massless fermions near the Fermi level, leading to unusual transport, optical, and magnetic responses. The landscape includes Dirac semimetals, Weyl semimetals, and nodal-line semimetals, each characterized by distinct patterns of band crossings and topological invariants. For readers of condensed matter physics alike, topological semimetals provide a bridge between abstract concepts from topology and tangible phenomena in real materials, often accessible through ARPES and quantum transport experiments.
In a topological semimetal, conduction and valence bands touch at discrete points (or along lines) in momentum space, and these touchings carry topological charges that render the crossings robust against small perturbations that respect the protecting symmetries. The most familiar realizations involve Weyl and Dirac fermions, which in the solid-state setting emerge as low-energy excitations near band crossings with linear dispersion. When time-reversal and inversion symmetries are present in a particular arrangement, a fourfold degenerate Dirac point can split into pairs of Weyl points with definite chirality if either symmetry is broken. The resulting Weyl semimetals exhibit a host of novel features, including Fermi-arc surface states that terminate at the projections of Weyl nodes, and chiral transport phenomena tied to the nontrivial Berry curvature in momentum space. For a detailed discussion of the foundational concepts, see Dirac semimetal, Weyl semimetal, and Berry curvature.
This topic sits at the intersection of symmetry, topology, and materials science. The notion of a topological invariant—an integer that cannot change under smooth deformations—underpins why certain band touchings are stable. For Weyl nodes, the Chern number acts as a monopole charge of Berry curvature in momentum space, effectively making each node a magnetic-like source or sink of Berry flux. In nodal-line semimetals, the touching occurs along a closed loop in momentum space, leading to extended regions of gapless excitations and characteristic surface states that can host nearly flat bands. The interplay of crystal symmetries, spin-orbit coupling, and crystal chemistry decides which of these possibilities is realized in a given material, and how robust the crossings remain against perturbations such as strain, disorder, or interactions.
Fundamental concepts
Band touchings and protection: Topological semimetals feature band crossings near the Fermi level. In many cases these crossings are protected by crystal symmetries (such as rotational or mirror symmetries) and/or by the topology of the Bloch wavefunctions, making the crossings robust against small perturbations that do not break the protecting symmetries. For a thorough discussion of how symmetries enforce or protect degeneracies, see crystal symmetry and topological invariant.
Weyl and Dirac fermions in solids: In a Weyl semimetal, pairs of Weyl nodes of opposite chirality appear in the Brillouin zone and act as sources and sinks of Berry curvature. The low-energy excitations near these nodes resemble Weyl fermions from high-energy physics, but with the crucial difference that crystal momentum and symmetry determine their distribution. Dirac semimetals host fourfold degenerate crossings that can be viewed as superpositions of Weyl fermions when certain symmetries are broken. See Weyl semimetal and Dirac semimetal for details.
Fermi-arc surface states: A hallmark of Weyl semimetals is the appearance of surface states that form open contours in momentum space, connecting projections of bulk Weyl nodes of opposite chirality. These Fermi arcs are a direct manifestation of the bulk-boundary correspondence in topological materials. See Fermi arc for a broader discussion.
Berry curvature and anomalous transport: The Berry curvature acts like a magnetic field in momentum space and profoundly influences semiclassical electron dynamics, producing anomalous velocities and unusual magnetotransport. The integrated Berry curvature across occupied states gives rise to topological responses that can be probed experimentally through transport and spectroscopy. See Berry curvature.
Classification and representative materials
Dirac semimetals: Systems with fourfold degenerate band crossings at discrete momenta, protected by a combination of time-reversal, inversion, and crystal symmetries. When a symmetry is broken, these Dirac points can split into Weyl points. Representative materials include engineered or naturally occurring candidates within the Dirac semimetal family.
Weyl semimetals: Characterized by pairs of Weyl nodes with opposite chirality. The separation of nodes in momentum space, the presence of surface Fermi arcs, and the chiral anomaly-driven transport are central features. The first experimentally confirmed Weyl semimetals were discovered in compounds such as TaAs and related pnictides, which sparked a large surge of activity in identifying and engineering Weyl phases.
Nodal-line semimetals: In these materials, the conduction and valence bands touch along one- or more closed loops in the Brillouin zone, rather than at isolated points. This topology leads to distinctive surface states and distinct responses to external fields. Notable examples include materials based on the ZrSiS family and related compounds.
In practice, researchers identify a material as a topological semimetal by combining theoretical band structure calculations with experimental probes like ARPES and quantum oscillations. For key material examples and discovery histories, see TaAs (a canonical Weyl semimetal), NbP and NbAs (related Weyl semimetals), and ZrSiS (a widely studied nodal-line semimetal). The field often relies on a dialogue between theory and experiment to confirm the presence and character of Weyl nodes, Dirac points, or nodal lines, as well as the associated surface states. See angle-resolved photoemission spectroscopy and scanning tunneling microscopy for complementary experimental approaches.
Experimental probes and phenomenology
Angle-resolved photoemission spectroscopy (ARPES): ARPES is the principal tool for directly imaging band dispersions, Weyl nodes, and Fermi arcs on the surface. It provides momentum-resolved information about the electronic structure that is crucial for identifying the topological character of a material. See ARPES.
Quantum oscillations and magnetotransport: Measurements of Shubnikov–de Haas and de Haas–van Alphen oscillations reveal details about the Fermi surface and effective masses, while large, non-saturating magnetoresistance and signatures of the chiral anomaly can point to Weyl physics in a material. See quantum oscillations and magnetoresistance.
Surface versus bulk sensitivity: The surface states (like Fermi arcs) are intimately tied to the bulk topology, but their experimental signatures can depend on surface termination and reconstruction. This makes careful surface preparation and interpretation essential. See surface states.
The role of interactions and disorder: Real materials host electron-electron interactions and imperfections that can modify or mask idealized band-structure pictures. Ongoing work assesses how robust the topological features remain under correlation effects and disorder, and whether interaction-driven phases emerge in proximity to topological semimetal phases. See electron correlation and disorder.
Theoretical framework
Topological invariants and Berry phase: The existence of protected band touchings is tied to topological invariants computed from Bloch states, such as Chern numbers and monopole charges of Berry curvature. This framework explains why certain crossings cannot be removed without a symmetry-breaking perturbation. See topological invariant and Berry phase.
Surface–bulk correspondence: The presence of surface states like Fermi arcs is rooted in the bulk topology. The bulk-band topology dictates the possible surface band structure, yielding characteristic experimental fingerprints. See bulk-boundary correspondence.
Model Hamiltonians: Minimal lattice models (tight-binding or k·p representations) capture the essential physics of Weyl and Dirac semimetals, illustrating how symmetry, spin-orbit coupling, and crystal structure shape the location and nature of band touchings. See tight-binding model and k dot p theory.
Controversies and debates
Material realizations and interpretation: While several compounds have been proposed as Weyl or Dirac semimetals, the certainty of their topological character can be debated when experiments yield ambiguous or surface-dependent results. Cross-checks via multiple probes and careful consideration of surface conditions are standard practice.
Role of interactions and disorder: The extent to which electron-electron interactions or impurities alter the idealized picture of massless fermions remains an active area of study. Some researchers argue for robust, interaction-protected features in certain regimes, while others emphasize that real materials may quickly depart from simple single-particle pictures under modest perturbations.
Classification boundaries: The taxonomy of semimetals—Weyl, Dirac, nodal-line, and higher-order variants—rests on both symmetry and topology. In some materials, experimental signatures may straddle categories or depend sensitively on external fields, strain, or subtle symmetry-breaking terms, prompting ongoing refinement of classification criteria.
Applications versus fundamental science: The excitement around potential devices and sensors must be balanced with the recognition that practical applications depend on finding materials with clean, tunable, and reproducible properties. Some critics emphasize fundamental science and reproducibility over hype, while others highlight the long-term payoff of discovering materials with extraordinary transport properties.
Applications and outlook
Topological semimetals offer a platform to test ideas from high-energy physics in solid-state settings, while their unusual transport properties hint at novel device concepts. Potential directions include low-dissipation electronics leveraging high carrier mobility, spintronic applications that exploit spin-momentum locking, and sensors that take advantage of large magnetoresistance or anomalous Hall-like responses. Ongoing materials discovery, improved synthesis techniques, and advances in spectroscopic and transport measurements are broadening the catalog of candidate compounds and refining the understanding of how topology, symmetry, and interactions conspire to shape electronic behavior.
The field maintains a strong interdisciplinary character, drawing from crystal structure design, first-principles calculation methods, and experimental techniques such as ARPES and STM to build coherent pictures of topological semimetal phases in real materials. As researchers map the relationships between crystallography, electronic structure, and observable phenomena, the catalogue of known semimetals continues to grow, enabling more precise tests of topological transport and surface-state physics.