凝聚態物理里的 Weyl 點的手性是什麼含義？如何判斷？從能帶上能看出來手性嗎？

01-08

Weyl points are singularities (conduction and valence bands touch) in band structure of odd dimensions. Near the Weyl point $k^w$ , the effective Hamiltonian is

$[H = sum_{j=1}^d v_j(k_j - k_j^w)sigma_j]$ with $v_j$ the Fermi velocity along different directions. Here $sigma_j$ are generators of an irreducible representation of the odd complex Clifford algebra $Cl_d$ .

The chirality $u_W$ is given by

$[ u_W = prod_{j=1}^d mathrm{Sgn}v_j]$ .

The chirality tells us whether the Weyl point is a source or a sink of Chern flux in the Brillouin zone.

Weyl points can only be annihilated in pairs, with positive and negative chirality.
The chirality of Weyl nodes in a Weyl semimetal is very similar to the moving directions (left/right) in a 1D chiral fermionic systems. In the presence of electric field, the occupation of isolated left/right branch is not conserved. There is a transfer of occupation from one branch to another, known as the chiral anomaly.
For a 3D Weyl semimetal, in the presence of EM field, the occupation at Weyl points with different chiralities will be incresed/decreased depdending on the chirality, this is the topological magnetoelectric effect. But if you sum over all Weyl nodes, the total occupation is conserved.