Weyl and Dirac Semimetals: Exotic Fermions in Crystals

Discover Weyl and Dirac semimetals — topological phases where quasiparticles mimic high-energy fermions, giving rise to Fermi arcs, chiral anomalies, and more.

Written by: Ajay Kumar

Posted: 7/15/2025

Weyl Semimetals Dirac Fermions Topological Materials
Weyl and Dirac cones in topological semimetals

Weyl and Dirac Semimetals: Exotic Fermions in Crystals

Introduction: Where High-Energy Physics Meets Condensed Matter

In the previous post, we explored the Quantum Spin Hall Effect — a 2D topological phase protected by spin and symmetry. Now, we ascend into a new realm where electrons inside crystals behave like relativistic particles from quantum field theory. Welcome to the world of Weyl and Dirac semimetals.

These topological phases don’t require an insulating gap. Instead, they host gapless points in their band structures — where energy bands cross linearly, creating quasiparticles that mimic Weyl or Dirac fermions. First predicted in high-energy physics, these fermions now emerge in the solid-state lab, complete with chiral anomalies, Fermi arcs, and topological transport phenomena.

Introduction to Weyl and Dirac Fermions

Dirac Fermions:

  • Originally described by the Dirac equation, these are massless, spin-½ particles that preserve both time-reversal and inversion symmetry.
  • In crystals, a Dirac point is formed by the overlap of two Weyl points with opposite chirality.

Weyl Fermions:

  • Predicted in 1929 but long considered unobservable, Weyl fermions emerge when either time-reversal or inversion symmetry is broken.
  • They act as chiral particles — having a handedness (left or right).
  • In condensed matter, Weyl nodes appear in pairs and act as monopoles of Berry curvature.

These fermions are not fundamental particles here, but quasiparticles — emergent excitations arising from the topological structure of electronic bands.

What Makes Weyl and Dirac Semimetals Unique?

Weyl and Dirac semimetals are not traditional metals or insulators. Their distinctive features include:

  • Linear dispersion in all three spatial dimensions near node points.
  • Topologically protected band crossings (Weyl/Dirac nodes).
  • Extremely high mobility of carriers due to linear dispersion.
  • Fermi arcs — surface states that connect Weyl nodes of opposite chirality.

Unlike conventional semimetals, their electronic behavior is dictated by topological invariants and Berry curvature, making them sensitive to magnetic fields, geometry, and symmetry.

Fermi Arcs and Chiral Anomaly

Fermi Arcs:

  • In Weyl semimetals, surface states form open-ended arcs in momentum space.
  • These arcs connect projections of bulk Weyl nodes of opposite chirality on the surface Brillouin zone.
  • This violates traditional expectations where Fermi surfaces must form closed loops.

Chiral Anomaly:

  • A quantum mechanical effect where chirality is not conserved in the presence of parallel electric and magnetic fields.
  • In Weyl semimetals, this results in negative magnetoresistance — conductivity increases with magnetic field.
  • It is a transport signature of the topological nature of the system.

These phenomena offer experimental fingerprints that distinguish Weyl and Dirac semimetals from ordinary materials.

Real Examples: TaAs, Na₃Bi

Several real-world materials have been confirmed to host these exotic quasiparticles:

Weyl Semimetals:

  • Tantalum arsenide (TaAs): The first Weyl semimetal observed experimentally.
    • Exhibits clear Fermi arcs via ARPES.
    • Demonstrates chiral anomaly in transport measurements.
  • NbAs, TaP: Similar behavior with strong spin-orbit coupling.

Dirac Semimetals:

  • Na₃Bi: A 3D Dirac semimetal protected by crystal symmetries.
  • Cd₃As₂: Known for ultrahigh mobility and large magnetoresistance.

These materials have band structures tuned by symmetry, allowing transitions between trivial, Dirac, and Weyl phases by breaking specific symmetries.

Experimental Techniques to Probe Weyl/Dirac Semimetals

Unveiling the topological features of these materials requires cutting-edge methods:

  • ARPES (Angle-Resolved Photoemission Spectroscopy): Directly maps band structures and surface states, revealing Fermi arcs.
  • Magnetotransport Measurements: Detect signatures of chiral anomaly through field-dependent conductivity.
  • Scanning Tunneling Microscopy (STM): Visualizes local density of states and quasiparticle interference.
  • Quantum Oscillation Experiments: Reveal Berry phase effects and topological signatures in Landau levels.

Together, these tools confirm the existence and stability of Weyl and Dirac nodes — and help identify new candidate materials.

Emerging Applications

While still early-stage, these materials are being explored for:

  • Topological electronics: Devices using robust, high-speed surface states.
  • Valleytronics: Utilizing the chirality of Weyl fermions as an information carrier.
  • Spin-based logic: Leverage spin-momentum locking for low-power computing.
  • Quantum sensing: High mobility and Berry curvature make these systems sensitive to magnetic field variations.

Their unusual physical properties make them attractive for quantum devices, next-gen transistors, and even gravitational wave analog studies.

Conclusion: Crystalline Portals to Quantum Relativity

Weyl and Dirac semimetals blur the lines between high-energy physics and materials science. They transform relativistic equations into tangible electronic states, unlocking rich quantum behaviors in tabletop experiments.

In the next post, we shift into a superconducting framework — exploring how topology combines with electron pairing to give rise to Majorana modes and topological superconductors.

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