10.2 Topological semimetals

Fundamentals of topological semimetals

Topological semimetals are quantum materials where conduction and valence bands cross at protected points or lines in momentum space. Unlike topological insulators (which have a bulk gap) or conventional metals (which have large Fermi surfaces), these materials sit in between: their band crossings create exotic quasiparticles and surface states that are robust against perturbation. This makes them a rich playground for studying fundamental quantum phenomena and a promising platform for future electronic devices.

Band structure characteristics

The defining feature of a topological semimetal is a nontrivial band crossing that occurs at discrete points or along lines in the Brillouin zone. Near these crossings, the energy dispersion is linear in momentum, meaning the low-energy electrons behave like massless relativistic fermions (Dirac or Weyl fermions) rather than the parabolic-dispersion electrons you'd find in a conventional semiconductor.

These band degeneracies are topologically protected: you cannot gap them out with small perturbations unless you break the symmetries that stabilize them. This is fundamentally different from an accidental degeneracy, which can be lifted by any generic perturbation.

Symmetry protection mechanisms

Symmetry is what keeps the band crossings stable. The relevant symmetries depend on the type of semimetal:

• Crystalline symmetries (inversion, rotation, mirror) protect band crossings in most topological semimetals
• Time-reversal symmetry is specifically required for Dirac semimetals, where it combines with crystal symmetry to enforce a four-fold degeneracy
• Breaking a protecting symmetry can gap out the crossing or split it into a different topological phase. For example, breaking either time-reversal or inversion symmetry in a Dirac semimetal can split each Dirac node into two Weyl nodes

This symmetry dependence also means you can engineer topological phase transitions by applying strain, magnetic fields, or chemical substitution to selectively break symmetries.

Bulk-boundary correspondence

Bulk-boundary correspondence is the principle that the topology of the bulk band structure guarantees the existence of specific surface states. In topological semimetals, this manifests most dramatically as Fermi arcs: open-ended contours on the surface Brillouin zone that connect the projections of bulk band-crossing points.

These surface states are topologically protected against backscattering from non-magnetic impurities, which makes them experimentally detectable and technologically interesting. The existence (or absence) of the predicted surface states serves as a direct experimental test of whether a material is truly a topological semimetal.

Types of topological semimetals

Topological semimetals are classified by the geometry and degeneracy of their band crossings. Each type has distinct symmetry requirements, surface states, and physical consequences.

Weyl semimetals

Weyl semimetals have isolated, two-fold degenerate band crossing points called Weyl nodes. Each Weyl node acts as a monopole (source or sink) of Berry curvature in momentum space, carrying a definite chirality of $+1$ or $-1$. Weyl nodes always come in pairs of opposite chirality (the Nielsen-Ninomiya theorem guarantees this on a lattice).

To realize Weyl nodes, you must break either time-reversal symmetry or inversion symmetry. If both are present, every band is at least two-fold degenerate everywhere, and you get Dirac nodes instead.

• Surface states: Fermi arcs connecting projections of Weyl nodes of opposite chirality
• Signature transport effect: Negative longitudinal magnetoresistance from the chiral anomaly
• Key materials: TaAs, NbAs, and magnetic Weyl semimetal $\text{Co}_3\text{Sn}_2\text{S}_2$

Dirac semimetals

Dirac semimetals have four-fold degenerate band crossings protected by the combination of time-reversal symmetry, inversion symmetry, and an additional crystal rotation symmetry. You can think of each Dirac node as two overlapping Weyl nodes of opposite chirality.

Because both Weyl copies are present at the same point, the net chirality is zero. Breaking time-reversal or inversion symmetry splits each Dirac point into a pair of Weyl points, providing a route to engineer Weyl semimetals from Dirac semimetals.

• The energy dispersion is linear in all three momentum directions near the Dirac point
• Key materials: $\text{Cd}_3\text{As}_2$, $\text{Na}_3\text{Bi}$

Nodal line semimetals

Instead of discrete points, nodal line semimetals have band crossings along continuous one-dimensional curves (closed loops or extended lines) in momentum space. These crossings are typically protected by mirror symmetry combined with time-reversal symmetry or other crystal symmetries.

• Surface states: "Drumhead" surface states, which are nearly flat bands filling the interior of the projected nodal loop on the surface Brillouin zone
• These flat drumhead states can enhance electron-electron correlation effects and lead to unusual instabilities
• Key materials: $\text{PbTaSe}_2$, $\text{ZrSiS}$

Topological invariants

Topological invariants are integer-valued (or $\mathbb{Z}_2$-valued) quantities computed from the band structure that remain unchanged under smooth deformations. They classify distinct topological phases and predict the existence of protected surface states.

Berry phase and curvature

When a quantum state is adiabatically transported around a closed loop in parameter space (here, momentum space), it picks up a geometric phase called the Berry phase:

$\gamma = \oint \mathbf{A}(\mathbf{k}) \cdot d\mathbf{k}$

where $\mathbf{A}(\mathbf{k}) = -i \langle u_{\mathbf{k}} | \nabla_{\mathbf{k}} | u_{\mathbf{k}} \rangle$ is the Berry connection. The Berry curvature $\mathbf{\Omega}(\mathbf{k}) = \nabla_{\mathbf{k}} \times \mathbf{A}(\mathbf{k})$ acts like an effective magnetic field in momentum space.

Berry curvature directly affects electron dynamics: it produces an anomalous velocity perpendicular to applied electric fields, which underlies the anomalous Hall effect. In Weyl semimetals, Berry curvature is concentrated around the Weyl nodes, which act as monopole sources and sinks.

Chern number

The Chern number is obtained by integrating Berry curvature over a closed 2D surface in momentum space:

$C = \frac{1}{2\pi} \oint \mathbf{\Omega} \cdot d\mathbf{S}$

This is always an integer. For a closed surface enclosing a single Weyl node, $C = \pm 1$, confirming the node's monopole character. The Chern number determines the number and chirality of chiral edge/surface modes, and a nonzero value guarantees topologically protected surface states.

$\mathbb{Z}_2$ invariant

The $\mathbb{Z}_2$ invariant classifies systems with time-reversal symmetry. It takes only two values:

• $\nu = 0$: trivial phase
• $\nu = 1$: topological phase

In 3D, there are four independent $\mathbb{Z}_2$ indices $(\nu_0; \nu_1 \nu_2 \nu_3)$. The strong index $\nu_0$ is the most important: if $\nu_0 = 1$, the system is a strong topological insulator with robust surface states on every surface.

For Dirac semimetals, the $\mathbb{Z}_2$ classification connects to the bulk topology through parity eigenvalues at time-reversal invariant momenta (the Fu-Kane criterion). The $\mathbb{Z}_2$ framework is less directly applicable to Weyl semimetals, which break time-reversal or inversion symmetry.

Fermi arcs and surface states

Fermi arcs are the most distinctive experimental signature of Weyl and Dirac semimetals. Unlike the closed Fermi contours found on surfaces of ordinary metals, Fermi arcs are open curves that terminate at the surface projections of bulk band-crossing nodes.

Formation of Fermi arcs

Fermi arcs arise directly from bulk-boundary correspondence. Here's the logic:

1. Each Weyl node is a monopole of Berry curvature with Chern number $\pm 1$
2. On any 2D slice of the Brillouin zone between two Weyl nodes of opposite chirality, the net Chern number is nonzero
3. A nonzero Chern number on a 2D slice mandates a chiral edge state on that slice
4. Collecting these edge states across all such slices produces a continuous arc on the surface Brillouin zone

The arc's length and shape depend on the momentum-space separation of the Weyl nodes and on surface-specific details (surface orientation, termination, and potential).

Experimental observations

• ARPES (angle-resolved photoemission spectroscopy) is the primary tool for directly imaging Fermi arcs. The first clear observations came from TaAs-family Weyl semimetals.
• Fermi arcs exhibit spin-momentum locking: the spin orientation rotates along the arc, determined by the chirality of the connected Weyl nodes
• In $\text{Cd}_3\text{As}_2$, surface states consistent with Dirac semimetal topology have also been resolved by ARPES
• Quasiparticle interference patterns measured by STM provide complementary momentum-space information about surface states

Surface state properties

Topological surface states are robust against scattering from non-magnetic disorder because backscattering would require connecting states of opposite spin/chirality, which is forbidden by the topological protection. This leads to:

• High surface mobility compared to trivial surface states
• Distinctive contributions to quantum oscillations in thin-film or nanostructure geometries (Weyl orbits connecting top and bottom surfaces through bulk Weyl nodes)
• Potential utility in spintronics, where spin-momentum locked surface currents could enable efficient spin-charge conversion

Transport properties

The nontrivial band topology of topological semimetals produces transport signatures that are qualitatively different from those of conventional metals. These serve both as experimental diagnostics and as the basis for potential device applications.

Chiral anomaly

The chiral anomaly is a quantum effect where chiral charge (the difference in electron population between Weyl nodes of opposite chirality) is not conserved when parallel electric and magnetic fields are applied. The mechanism works as follows:

1. Apply $\mathbf{E} \parallel \mathbf{B}$ along the direction connecting Weyl nodes in momentum space
2. The zeroth Landau level at each Weyl node disperses in one direction only (determined by chirality)
3. Electrons are pumped from one Weyl node to the other, creating a chiral charge imbalance
4. This imbalance relaxes through inter-node scattering on a timescale $\tau_v$, establishing a steady-state nonequilibrium population

The net result is an enhanced conductivity along the field direction, since the pumping creates additional current-carrying states.

Negative magnetoresistance

The chiral anomaly produces a negative longitudinal magnetoresistance (NMR): resistance decreases as the magnetic field increases when $\mathbf{E} \parallel \mathbf{B}$. This is unusual because in most metals, magnetoresistance is positive.

Key experimental signatures to distinguish genuine chiral-anomaly NMR from artifacts:

• The effect is maximized when $\mathbf{E}$ and $\mathbf{B}$ are exactly parallel and vanishes as they become perpendicular
• The angular dependence follows a $\cos^2\theta$ pattern (where $\theta$ is the angle between $\mathbf{E}$ and $\mathbf{B}$)
• Current jetting effects in high-mobility samples can mimic NMR, so careful contact geometry is essential

Quantum oscillations

Quantum oscillations (Shubnikov-de Haas in resistivity, de Haas-van Alphen in magnetization) are periodic in $1/B$ and reveal the extremal cross-sectional areas of the Fermi surface. In topological semimetals, they carry additional information:

• The phase offset of the oscillations encodes the Berry phase accumulated around the cyclotron orbit. A Berry phase of $\pi$ (expected for a Dirac/Weyl dispersion) shifts the Landau level fan diagram by $1/2$
• Unusual angular dependence of oscillation frequencies can reveal the anisotropy and topology of the Fermi surface near Weyl/Dirac points
• In thin samples, "Weyl orbits" that traverse the bulk via Weyl nodes and connect top and bottom Fermi arcs produce oscillation frequencies dependent on sample thickness

Experimental techniques

Confirming that a material is a topological semimetal requires converging evidence from multiple probes: one technique maps the band structure, another measures transport, and a third provides real-space surface information.

ARPES for band structure

Angle-resolved photoemission spectroscopy measures the energy and momentum of photoelectrons ejected from a sample surface, directly mapping $E(\mathbf{k})$.

What ARPES reveals in topological semimetals:

• Linear band crossings at or near the Fermi level, confirming Dirac/Weyl dispersion
• Fermi arc surface states and their connectivity to bulk node projections
• Spin texture via spin-resolved ARPES, verifying spin-momentum locking

Limitations: ARPES is surface-sensitive (probing depth of a few angstroms), requires clean surfaces (typically cleaved in ultra-high vacuum), and can only access occupied states below the Fermi level. Pump-probe (time-resolved) ARPES can partially access unoccupied states.

Magnetotransport measurements

Magnetotransport probes the bulk electronic response and is the primary method for detecting chiral anomaly effects.

• Longitudinal magnetoresistance ($\mathbf{I} \parallel \mathbf{B}$): look for negative MR as a chiral anomaly signature
• Hall effect: anomalous Hall contributions from Berry curvature
• Quantum oscillations: extract Fermi surface geometry and Berry phase from the oscillation pattern
• Planar Hall effect: an additional signature of the chiral anomaly, where a transverse voltage appears even for in-plane fields

Scanning tunneling spectroscopy

STM/STS provides real-space maps of the local density of states with atomic resolution.

• Quasiparticle interference (QPI) imaging: Fourier-transforming real-space conductance maps reveals the allowed scattering wavevectors, which encode information about the surface-state dispersion and any forbidden backscattering channels
• Local spectroscopy near defects tests whether surface states are truly immune to backscattering
• Can probe both occupied and unoccupied states (unlike ARPES), giving a more complete picture of the local electronic structure

Materials and realizations

Inorganic crystalline materials

Most confirmed topological semimetals are inorganic single crystals:

• Weyl semimetals: The TaAs family (TaAs, TaP, NbAs, NbP) breaks inversion symmetry while preserving time-reversal symmetry. Magnetic Weyl semimetals like $\text{Co}_3\text{Sn}_2\text{S}_2$ break time-reversal symmetry instead.
• Dirac semimetals: $\text{Cd}_3\text{As}_2$ and $\text{Na}_3\text{Bi}$ are the best-studied examples, with Dirac points protected by rotational crystal symmetry along specific axes.
• Nodal line semimetals: $\text{ZrSiS}$ and related compounds host nodal lines protected by nonsymmorphic crystal symmetries; $\text{PbTaSe}_2$ is another well-characterized example.

Engineered heterostructures

Topological semimetal phases can also be created artificially:

• Superlattices of alternating topological insulator and normal insulator layers can produce Weyl semimetal states when the layer thicknesses are tuned to close and reopen the bulk gap
• Strain engineering at interfaces can break symmetries in a controlled way, driving transitions between Dirac, Weyl, and trivial phases
• Heterostructure approaches allow tuning of Weyl node separation and Fermi arc length through composition and layer thickness

Photonic and acoustic analogs

Topological band theory is not limited to electrons. Photonic crystals and acoustic metamaterials can be designed to have band structures with Weyl points, nodal lines, and topological surface states.

• These classical analogs provide macroscopic, directly measurable platforms for studying topological physics
• Weyl points have been realized in 3D photonic crystals with broken inversion symmetry, and the resulting surface states act as one-way waveguides
• Acoustic topological semimetal analogs demonstrate robust sound propagation along surface channels, immune to defects

Applications and future prospects

Spintronics and quantum computing

• Spin-momentum locked surface states can convert charge currents to spin currents with high efficiency, useful for spin-orbit torque devices
• Topological protection could improve qubit coherence times if topological semimetal heterostructures can host Majorana zero modes at interfaces with superconductors
• Proximity-induced superconductivity in Weyl/Dirac semimetals is an active area of research aimed at realizing topological superconducting states

High-performance electronics

• The linear dispersion and high carrier mobility in materials like $\text{Cd}_3\text{As}_2$ (mobilities exceeding $10^6 \, \text{cm}^2/\text{V·s}$) make them candidates for ultra-fast transistors and detectors
• Chiral-anomaly-based magnetoresistive effects could enable new types of magnetic field sensors
• The broadband optical response from gapless linear dispersion is promising for terahertz detection and emission

Topological quantum chemistry

Topological quantum chemistry (TQC) is a framework that systematically classifies band topology using the language of crystal symmetry groups and band representations.

• TQC compares the symmetry representations of occupied bands at high-symmetry momenta against those allowed for atomic-limit (trivially localized) states
• A mismatch indicates nontrivial topology, enabling automated, high-throughput screening of materials databases
• This approach has identified thousands of candidate topological materials, dramatically expanding the known landscape of topological semimetals and insulators
• Ongoing work explores how strong electron-electron correlations interact with band topology, potentially producing new phases not captured by single-particle band theory