So far we’ve seen two examples of Dirac points in two dimensions, the surface of a 3D topological insulator and graphene. You might wonder, why don’t we have such cones in three dimensions? These do indeed exist and are called Weyl points instead of Dirac points. The reason is historical - Dirac’s equation for the electron (which is in 3D) involves states with four components, two for the electron and two for the hole. The direct generalization of graphene to \(3D\) that we will discuss involves states with two electron component. Such electron states with linear dispersion were studied first by Weyl, and have strange properties as we will illustrate below.

Let us start by writing the low-energy Hamiltonian for the three dimensional generalization of graphene:

\[H({\bf k})=(\sigma_x k_x+\sigma_y k_y+\sigma_z k_z).\]

Here you might think of \(\sigma_{x,y,z}\) as the spin of the electron (just as on the surface of a topological insulator).

Next we try the usual thing we would do with a two-dimensional Dirac cone - namely see what happens when we gap it out by applying a magnetic field \(\bf\sigma\cdot B\). Adding such a term, we find that the Hamiltonian transforms as follows:

\[H({\bf k})\rightarrow H({\bf k})+{\bf\sigma\cdot B}={\bf\sigma\cdot (k+B)}.\]

The key observation here is that the addition of a magnetic field effectively shifts the wave-vector as

\[{\bf k}\rightarrow \tilde{\bf k}={\bf k+ B}.\]

So applying the most general perturbation we can think of does not gap out the Weyl point where the energy vanishes. Instead, the perturbation only shifts the Weyl point around in momentum space. This feels like some kind of topological protection.

Is there a sense in which Weyl points are “topological”? They are clearly protected, but is there some topological reason for the protection? As in the rest of this section, the topology of gapless system becomes apparent by looking at the Hamiltonian in lower dimensional subspaces of momentum space. For the case of Weyl, the momentum space is three dimensional, so let us look at two dimensional subspaces of momentum space.

A natural subspace to choose is to fix \(k_z=m\). The Weyl Hamiltonian then becomes that of a massive 2D Dirac cone

\[H_{2D,Dirac}(k_x,k_y;m)\equiv H(k_x,k_y,k_z=m)=(\sigma_x k_x+\sigma_y k_y+m\sigma_z).\]

As we talked about in week 4 with Chern insulators, the massive Dirac model has a Chern number, which changes by \(1\) if \(m\) changes sign.

So we can think of the Weyl Hamiltonian in the momentum planes at fixed \(k_z\) as Chern insulators with Chern numbers \(n_{Ch}=0\) (i.e. trivial) if \(k_z < 0\) and \(n_{Ch}=1\) (topological) if \(k_z > 0\). The Hamiltonian at \(k_z=0\) is at the phase transition point of the Chern insulator, which supports a gapless Dirac point.

Systems with Weyl points are known as Weyl semimetals. Just like other topological phases, Weyl semimetals have an interesting surface spectrum. We can understand this easily by viewing the Weyl point as a stack of Chern insulators in momentum space. For any surface in a plane that contains the \(z\)-axis, we can treat \(k_z\) as a conserved quantity. At this \(k_z=m\), the Hamiltonian is just that of a Chern insulator with an appropriate Chern number. For the range of \(k_z\) where the Chern number \(n_{Ch}(k_z)=1\), the surface spectrum supports chiral edge states with an energy approximated at low energy by

\[E(k_x,k_z)\approx v(k_z)k_x.\]

We can consider the edge states over a range of \(k_z\) together to visualize the “surface states”.

The unique property of the surface states is that if we set \(k_x=0\) then the energy vanishes on a line in the surface spectrum. This line actually terminates at \(k_z=0\), where the Chern number changes. Such lines, which are referred to as “Fermi arcs,” are the unique bounday properties (hence the bulk-boundary correspondence) of Weyl semimetals.

At large enough \(k_z\), the two dimensional Hamiltonian \(H_{2D,Dirac}(k_x,k_y;k_z)\) becomes trivial i.e. \(n_{Ch}(|k_z|\rightarrow \infty)=0\). This means that if the Chern number is \(n_{Ch}=1\) in a range of \(k_z\), then \(n_{Ch}(k_z)\) must change twice resulting in two Weyl points. So Weyl points come in pairs. These points map onto the ends of the Fermi arcs on the surface.

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