Weyl Semimetal Band Structure and Fermi Arcs

Recommended resources

The freely available resources this lesson will direct you to.

Hello! Welcome to the next lesson in our exploration of topological matter.

In our previous lesson, we established the bulk-boundary correspondence for topological insulators, understanding that a topologically non-trivial gapped bulk mandates the existence of gapless states at the boundary. Today, we venture into a different realm of topological materials: topological semimetals. These materials are fascinating because their bulk is gapless, but only at specific, topologically protected points in momentum space.

Our learning outcome for this lesson is to describe the bulk band structure and surface Fermi arcs characteristic of a Weyl semimetal. We will discover that the topological nature of the bulk band touchings leads to one of the most exotic electronic structures in condensed matter: open, disconnected Fermi surfaces on the crystal's boundary.

1. From Dirac to Weyl Fermions in Solids

Let's start by recalling the critical point of a 3D topological phase transition. As we saw, the transition between a trivial insulator and a topological insulator occurs when the bulk band gap closes and reopens. Right at this critical point, the material is a gapless semimetal, and the low-energy excitations can be described by the 3D Dirac equation. Such a material is called a Dirac semimetal.

A massless 3D Dirac fermion is a four-component object, which can be thought of as being composed of two degenerate, two-component Weyl fermions with opposite chirality. Chirality is a "handedness" associated with a massless particle, indicating whether its spin is aligned or anti-aligned with its momentum.

In a Dirac semimetal, these two Weyl fermions are located at the same point in momentum space. The key to creating a Weyl semimetal is to separate them. This can be achieved by breaking either time-reversal symmetry (TRS) or inversion symmetry (IS).

Breaking TRS (e.g., with magnetic ordering) splits the Dirac point into two Weyl points separated in momentum space.
Breaking IS (in a non-centrosymmetric crystal) also splits the Dirac point. Due to the remaining TRS, if there's a Weyl point at momentum $k$ , there must be another one at $- k$ with the same chirality. The Nielsen-Ninomiya theorem (which we'll discuss shortly) then requires another pair at $k^{'}$ and $- k^{'}$ with opposite chirality, leading to a minimum of four Weyl points.

To see this formally, let's study a concise derivation.

Week 10 Lecture Notes: Topological Condensed Matter

These lecture notes from ETH Zurich provide a clear, mathematical derivation of a Weyl semimetal. They begin with the Hamiltonian for a 3D topological insulator at its critical point (a Dirac semimetal) and show how breaking time-reversal symmetry with a magnetic field splits the Dirac point into two distinct Weyl points.

Please read section 10.1, starting from the beginning and ending just before the discussion of the Berry curvature (before Eq. 10.1.8). Focus on how the Hamiltonian in Eq. (10.1.6) describes two separate Weyl points.

As the notes show, the low-energy Hamiltonian near a single Weyl point takes the beautifully simple form:
$H_{χ} (k) = χ v_{F} σ \cdot k$
Here, $k$ is the momentum measured from the node, $σ$ is the vector of Pauli matrices, $v_{F}$ is the Fermi velocity, and $χ = \pm 1$ is the chirality, or topological charge, of the node. This Hamiltonian describes a linear crossing of two bands—a 3D analogue of graphene's Dirac cone, but with only two bands instead of four.

2. Bulk Band Structure: Weyl Points as Monopoles of Berry Curvature

The most profound property of a Weyl point is its topological nature. In the previous module, you learned that the Chern number is the integral of the Berry curvature over the 2D Brillouin zone. A Weyl point acts as a source or sink of Berry curvature in 3D momentum space.

Specifically, a Weyl point is a monopole of Berry curvature.

The Berry curvature for a Weyl point of chirality $χ$ is given by:
$F (k) = χ \frac{k}{2∣ k ∣ ^{3}}$
This has the exact form of the magnetic field from a magnetic monopole at the origin! The chirality $χ$ is the "monopole charge".

If we integrate the Berry flux over any closed 2D surface $S$ enclosing a Weyl point in momentum space, we get a quantized Chern number equal to its chirality:
$C = \frac{1}{2 π} \oint_{S} F \cdot d S = χ = \pm 1$
A famous result, the Nielsen-Ninomiya theorem, states that the total topological charge within the entire Brillouin zone (which is a closed manifold, a 3-torus) must be zero. This has a powerful consequence: Weyl points must always appear in pairs of opposite chirality. You can create a pair of $+ 1$ and $- 1$ nodes, and you can move them around in momentum space, but you can only annihilate them by bringing them together. This is the origin of their topological protection.

3. Surface States: Fermi Arcs

Now we arrive at the heart of the bulk-boundary correspondence for Weyl semimetals. What do the surface states look like?

Consider a simple Weyl semimetal with two nodes of opposite chirality separated along the $k_{z}$ axis. We can think of the 3D Brillouin zone as a continuous stack of 2D Brillouin zones, parameterized by $k_{z}$ .

For $k_{z}$ values outside the region between the two Weyl points, a 2D slice of the bulk is fully gapped and topologically trivial (Chern number $C = 0$ ).
For $k_{z}$ values between the two Weyl points, the 2D slice intersects the "monopole field" of one Weyl point but not the other. This 2D subsystem is effectively a Chern insulator with $C = χ$ .

From our previous lesson, we know that a 2D system with a non-zero Chern number must host chiral edge states. Therefore, for every $k_{z}$ between the Weyl points, our material's surface must have a gapless state.

When we assemble all these 1D edge states from each $k_{z}$ slice, they form a 2D surface band. The Fermi surface of this band is not a closed loop, as is typical for 2D metals. Instead, it forms an open line segment that connects the projections of the two bulk Weyl points onto the surface Brillouin zone. This is the Fermi arc.

Bulk Band Structure and Surface Fermi Arcs of a Weyl Semimetal — This figure illustrates the key features of a Weyl semimetal. The bulk bands touch at two points, the Weyl nodes (W+ and W-), which have opposite chirality. On the surface, a unique state called a 'Fermi Arc' emerges, connecting the projections of these two bulk nodes. This is a direct manifestation of the bulk-boundary correspondence in a gapless system.

Week 10 Lecture Notes: Topological Condensed Matter

Let's revisit the ETH lecture notes to see a concise explanation of how Fermi arcs emerge.

Please read the two paragraphs on page 4 under the heading 'Another important property of Weyl semimetals are Fermi arcs...'. Also, examine Figure 10.1 and its caption, which provide an excellent visual summary of this concept.

A Fermi arc is a truly bizarre object. An electron moving along it in momentum space would abruptly vanish at one end (the projection of a Weyl point) and, to complete a circuit, would have to travel through the bulk to the other Weyl point before reappearing on the surface. This intimate connection between the bulk nodes and the surface arcs is the hallmark of a Weyl semimetal.

Test your understanding!

A Dirac semimetal has both time-reversal and inversion symmetry. Its bulk consists of Dirac points where four bands touch. Why don't Dirac semimetals have Fermi arcs connecting these points on their surface? (Hint: Think about the topological charge of a Dirac point).

Show answer

A Dirac point can be viewed as two Weyl points of opposite chirality sitting at the exact same point in momentum space. Therefore, the net topological charge of a Dirac point is $+ 1 + (- 1) = 0$ . Since the point itself has no net monopole charge, any 2D surface enclosing it in momentum space will have a Chern number of zero. Consequently, there is no topological reason for protected surface states like Fermi arcs to exist. The surface of a Dirac semimetal can be gapped, although some materials (like Na₃Bi) host surface states protected by crystalline symmetries.

4. Experimental Signatures in ARPES

The definitive experimental technique for observing Weyl semimetals is Angle-Resolved Photoemission Spectroscopy (ARPES), a method you'll be familiar with from your work with photoluminescence. ARPES directly measures the electron energy vs. momentum, $E (k)$ , effectively mapping the band structure.

A key feature of ARPES is that the escape depth of photoelectrons depends on their kinetic energy, which is controlled by the incident photon energy.

Low-energy photons (soft X-rays, UV) create low-energy photoelectrons that can only escape from the very top atomic layers. This makes the measurement surface-sensitive.
High-energy photons (hard X-rays) create high-energy photoelectrons that can escape from deep within the crystal. This makes the measurement bulk-sensitive.

This tunability is precisely what's needed to disentangle the bulk Weyl points from the surface Fermi arcs. The following video by Prof. Zahid Hasan, a pioneer in the experimental discovery of these materials, brilliantly illustrates this.

Prof. Zahid Hasan, "Weyl Fermions & Topological Fermi Arcs"

In this first segment, Prof. Hasan explains how ARPES is used to probe the bulk of a Weyl semimetal candidate. By using high-energy photons, the experiment becomes sensitive to the bulk band structure, revealing the Weyl points.

Please watch from 34:32 to 43:58. Focus on the distinction between low- and high-photon energy measurements. Notice how at high energy (bulk probe), the complex surface features disappear, and what remains are simple 'dots' in momentum space, which are the signatures of the bulk Weyl points.

Now for the crucial step: connecting the bulk to the surface.

Prof. Zahid Hasan, "Weyl Fermions & Topological Fermi Arcs"

This is the climax of the experimental discovery. Prof. Hasan overlays the surface-sensitive data (showing arcs) with the bulk-sensitive data (showing dots) to demonstrate the bulk-boundary correspondence directly.

Watch from 43:58 to 49:36. The key moment is when the two separate datasets are combined. Observe how the Fermi arcs, seen with low-energy photons, are shown to terminate exactly at the locations of the bulk Weyl points, seen with high-energy photons. This is the definitive experimental proof.

This experimental confirmation is a triumph of condensed matter physics, providing a stunning visualization of an abstract topological concept. It perfectly demonstrates how the properties of the infinite, periodic bulk dictate the physics on the finite surface of a crystal.

Conclusion

In this lesson, we have moved from gapped topological insulators to gapless topological semimetals, focusing on the Weyl semimetal as the prototypical example.

Key Takeaways:

Bulk Band Structure: A Weyl semimetal features points in its bulk Brillouin zone where the conduction and valence bands touch linearly. These are called Weyl points.
Topological Nature: Weyl points are topologically protected because they act as monopoles of Berry curvature in momentum space. They carry a quantized chiral charge ( $\pm 1$ ) and must appear in pairs of opposite charge.
Surface States: The bulk-boundary correspondence guarantees the existence of exotic surface states called Fermi arcs, which are open segments of a Fermi surface.
Bulk-Boundary Connection: The Fermi arcs connect the projections of the bulk Weyl points of opposite chirality onto the surface Brillouin zone. This connection is directly observable using ARPES.

Preview of the Next Lesson

Now that we have a solid understanding of the properties of a Weyl semimetal, we are equipped to make a more detailed comparison. In the next lesson, we will contrast the stability and properties of Weyl and Dirac semimetals, delving deeper into the role of symmetry and the distinct physical consequences of these two fascinating topological phases.