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.

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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


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?


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


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