Band structure of Weyl semimetal?

Question

Weyl semimetals are topological quantum materials whose low energy excitations emerges as massles Weyl Fermions. They have a band touching point near the Fermi level called Weyl node.

What is actually the energy dispersion relation of Weyl Fermions? More specifically, do we have to consider the band as a linear dispersion relation or as a conduction and valence band which takes the shape of two cones?

I came up with this doubt because if we consider the band dispersion relation as linear, then

$$E_k=C\times (k-k_0)$$(For convenience consider 1D. but I know that Weyl semimetals are three dimensional)

then the effective mass is given by $$m*=\frac{\hbar^2}{\frac{\partial^2E_k}{\partial k^2}}$$

that gives

$$m*=\frac{\hbar^2}{0}=\infty$$

Which contradicts the fact that Weyl Fermions are massless.

But If we consider it as the latter one then the derivative is not defined at the Weyl node? Where am I wrong in my calculations and concepts?

Answer 1

A Weyl point does indeed have linear dispersion. The effective mass tensor you are using $\frac{1}{m^*_{ij}}\propto\frac{\partial^2\varepsilon(k)}{\partial k_i \partial k_j}$ is not really well-defined here. After all, we only define the effective mass that way to match the common non-relativistic equation $\varepsilon = \frac{\hbar^2 k^2}{2m^*}$. A Weyl point is in no sense described by such an equation since its low-energy effective Hamiltonian is relativistic. Specifically, the effective Hamiltonian is a generalization of the Weyl Hamiltonian from high energy physics. This generalized form is $$ \tag{1}H(k) = (a_\mu\cdot k)\sigma^\mu $$ (more on this in the next section). The ordinary Weyl equation from high energy physics is given by the special case $H_{weyl}=k\cdot \sigma$. This describes massless chiral fermions as described in many other places.

General Weyl Point Dispersion

Now I will sketch the derivation of the general Weyl point dispersion (both Type-I and Type-II). In general, a Weyl point is an isolated linear crossing of two bands. In the vicinity of that crossing, we can focus on just those two bands which are described by a two-band Hamiltonian which is generally of the form $$\tag{2}H(k) = h_\mu(k)\sigma^\mu = h_0(k) \sigma^0 + h_x(k) \sigma^x + h_y(k)\sigma^y + h_z(k)\sigma^z$$ Taking the degenerate point to be at $k=0$ and at $\varepsilon=0$, we can expand the functions $h_\mu$ as $h_\mu(k) = \nabla_k h_\mu\big|_0 \cdot k + \dots$ so that the effective Hamiltonian at the Weyl point is $$ H(k) = (a_\mu\cdot k)\sigma^\mu $$ where $a_\mu = \nabla_k h_\mu\big|_0$ are model-dependent vector parameters. This can be diagonalized to arrive at the general Weyl point dispersion relation: $$ \tag{3}\varepsilon_\pm(k) = a_0\cdot k \pm \sqrt{(a_x\cdot k)^2 + (a_y\cdot k)^2 + (a_z\cdot k)^2}$$ Depending on $a_0$, this can be either a Type-I or Type-II Weyl point.

Multi-Weyl points

As a final note, I just want to mention that there are also so-called multi-Weyl points which have non-linear dispersion and have higher chiralities. More about them here