I started studying Weyl physics in condensed matters, but I got confusing about the difference between the Weyl point and Weyl node. I understood that when the Weyl points connect continuously, the Weyl node is created. Is it right?

In addition, is there any difference about the condition to realize Weyl point and node?


I think Weyl point and Weyl node are synonyms.

As to the condition for the existence of a Weyl point, then there are a few things to note. First, you need two bands crossing, so you need to break either time reversal symmetry or inversion symmetry, as otherwise all bands are doubly degenerate, so crossing points will involve four bands (rather than two). Second, consider a general two-band Hamiltonian near a crossing point in a 3-dimensional system:

$$ \hat{H}=\mathbf{q}\cdot\mathbf{\sigma}=q_1\sigma_1+q_2\sigma_2+q_3\sigma_3, $$

where $\mathbf{q}=(q_1,q_2,q_3)$ are the coordinates from the crossing point at $\mathbf{q}=\mathbf{0}$, and $\sigma=(\sigma_1,\sigma_2,\sigma_3)$ are the Pauli matrices. You can confirm that $\mathbf{q}=\mathbf{0}$ is the crossing point by calculating the dispersion:

$$ E_{\pm}=\pm\sqrt{q_1^2+q_2^2+q_3^2}. $$

Next, imagine adding a perturbation to this Hamiltonian:

$$ \hat{V}=\mathbf{m}\cdot\mathbf{\sigma}=m_1\sigma_1+m_2\sigma_2+m_3\sigma_3, $$

and you will end up with a new Hamiltonian:

$$ \hat{H}+\hat{V}=\mathbf{q}\cdot\mathbf{\sigma}+\mathbf{m}\cdot\mathbf{\sigma}=(q_1+m_1)\sigma_1+(q_2+m_2)\sigma_2+(q_3+m_3)\sigma_3, $$

whose dispersion is $E_{\pm}=\pm\sqrt{(q_1+m_1)^2+(q_2+m_2)^2+(q_3+m_3)^2}$. Therefore, you still have a Weyl point, it has simply moved to $\mathbf{q}=(-m_1,-m_2,-m_3)$. This means that two-band crossings are rather robust in 3 dimensions, and indeed materials that obey the right conditions (e.g. time reversal symmetry breaking) have them all over their band structures, see for example this paper. However, for these Weyl points to be interesting what you need is they sit near the Fermi energy, so that one of the touching bands is occupied and the other unoccupied, and this is not so easy to find. This argument is completely different in two dimensions. The Hamiltonian of a two-band crossing is now:

$$ \hat{H}=\mathbf{q}\cdot\mathbf{\sigma}=q_1\sigma_1+q_2\sigma_2, $$

because $\mathbf{q}$ is a 2-dimensional vector. Therefore, adding a perturbation like $\hat{V}$ will in general gap the system and remove the Weyl point. You will need extra symmetries to protect the crossing in 2 dimensions.

  • $\begingroup$ Thank you. I understood. $\endgroup$ – Y. P Jul 12 '20 at 6:02

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