What is the approx. separation of chiral Weyl nodes in recently discovered Chiral anomaly in Dirac semi-metals?

Question

Recent papers have shown evidence for the existence of the long sought after Chiral axial anomaly to be present in certain Weyl semi-metals.... Usually they talk of the parallel B and E fields inducing violation of symmetry so as to create oppositely chiral "Weyl nodes" (that results the "chiral magnetic effect" in violation of conserved current).....and sometimes they show evidence of Fermi arcs . But in all the reports I have seen I have never seen any indication of the magnitude of the spatial separation of the nodes....or even the number of nodes per amount of volume of material, etc. .... Usually only comments about the phenomenology between two nodes....

So 1st...what is a "node"? Isn't it the place of accumulation of a number of chiral "quasi-particles"??... and 2. What is the typical spatial separation of two nodes and what determines the amount of that separation? (I mean are we talking microns or inches?? ) ....or am I missing the whole idea due to my ignorance of solid state physics.? Thanks for the help.

Answer 1

1) The Weyl node is the point at which the two energy zones of the semimetal - the valence and the conductivity zones - are intersected. The particles spectrum in corresponding Brilloiun zone is continuously deformed from the standard Schroedinger one (in general the spectrum is determined from the material lattice structure) to the linear one, which is the Weyl fermion spectrum. So, the Weyl node contains effective massless Weyl fermions. Here You need to learn some details.

• Why two energy zones intersect each other only at points? The answer is von Neumann-Wigner crossing. Suppose 2 level hamiltonian $H$, and suppose we require that it has non-degenerate eigenvalues $E_{1}(c),E_{2}(c)$, where $\{ c\}$ is the set of parameters. Suppose next the levels are intersected along the some line $\gamma(c)$ in parameters space. The system tends to levels repulsion by generating the perturbation $\delta H$, which can be given in the general form $\delta H = \mathbf a \cdot \mathbf \sigma$, where $\mathbf a$ is the perturbation and $\sigma$ is the set of three Pauli matrices. In order to set $\mathbf a$ to zero we have to tune at least 3 free parameters $c$. In 3D space the only these parameters are the lattice momentum $\mathbf p$ components. By tuning 3 parameters we fix the given point $\mathbf p_{W}$, which is called the Weyl node. Only at this points two levels can intersect each other;
• Why the WEYL node? Near the given point the Hamiltonian $H$ (with the energy shift $H_{0}$) can be expanded as $$ \tag 1 H - H_{0}^{\lambda} = \lambda v_{F}(\mathbf p-\mathbf p^{\lambda}_{W}) \cdot \sigma , $$ where $v_{F}<1$ is the Fermi velocity, and $\lambda = \pm 1$ is what is called the effective chirality in solid body. The Hamiltonian $(1)$ has the form of Weyl hamiltonian, describing the massless fermions of given chirality $\lambda$ propagating with the velocity $v_{F}$.

2) The non-zero separation between the given pair of the Weyl nodes are rather in energy-momentum space. This is preferable since when we discuss the solid body lattice we typically make the Fourier transformation, for time as well as for spatial coordinates. The presence of this separation depends on whether the parity and time reversal symmetries is broken in semimetal lattice or not. This creates two classes of semimetals - Dirac or Weyl.

• Which nodes are separated? In fact, if there exist the node with given chirality $\lambda$ and given momentum $\mathbf p_{W}^{\lambda}$, then there must exist the another node with the chirality $-\lambda$ and momentum $\mathbf p^{-\lambda}_{W}$. This statement is known as Nielsen-Ninomiya theorem. It can be argued in a following way. Each Weyl node is the source for the so-called Berry curvature - the monopole-like field in momentum space, with the charge determined by the chirality $\lambda$. Its field lines must be ended somewhere in the Brillouine zone. The only way to do this is to require that there must be the node with an opposite chirality $-\lambda$. These two nodes - with chirality $\lambda$ and $-\lambda$ - may be separated by the finite distances in momentum and energy space, given by $$ \tag 2 \mathbf b \equiv \mathbf p_{W}^{\lambda}- \mathbf p_{W}^{-\lambda}, \quad b_{0} \equiv H_{0}^{\lambda} - H_{0}^{-\lambda} $$ So, there is even number of nodes, and the number of pairs is determined by the number of semimetal energy zones intersections.

• What determines the distance between Weyl nodes? In fact, it is determined by the parity and time reversal symmetries on a semimetal lattice. This can be easily see from the explicit form of the hamiltonian $H$, given by $(1)$ and $(2)$: $$ H(\mathbf b, b_{0}) = \begin{pmatrix} b_{0} + \sigma \cdot (\mathbf p - \mathbf b)& 0 \\ 0 & -b_{0}-\sigma \cdot (\mathbf p + \mathbf b)\end{pmatrix} $$ Under parity transformation, $H$ is transformed into $H(\mathbf b, -b_{0})$, while under time reversal transformation, $H$ is transformed into $H(-\mathbf b , b_{0})$. So that we see that if $\mathbf b \neq 0$, then the time reversal symmetry is broken, while if $b_{0} \neq 0$, then the parity symmetry is broken. If at least one of quantities $b_{0},\mathbf b$ is non-zero, then the semimetal is called the Weyl semimetal. The case $b_{0} = 0,\ \mathbf b = 0$ is called the Dirac semimetal.

3) Finally, since the Weyl fermion theory in presence of external electric and magnetic fields is anomalous, then the full effective theory of semimetal, desctibing the pairs of Weyl nodes, generate anomalous transport phenomena - the anomalous Hall effect and the chiral magnetic effect.