Figure 2

Comparison between Weyl, Dirac, and nexus band crossings. (a) A Weyl node arises from the crossing between two singly degenerate bands. The valence and conduction bands are noted as “1” and “2”. (b,c) We enclose the Weyl node by a sphere in k space. We notice that the sphere satisfies following two crucial conditions: (1) The sphere is a 2D closed manifold; (2) Bands 1 and 2 are separated by a band gap at all k points on the sphere. These two facts guarantee that one can calculate the Chern number of the filled valence bands on this sphere. Because the Weyl nodes are Berry curvature monopoles, it has been shown¹ the Chern number (C) of the sphere equals the chiral charge (χ) of the enclosed Weyl node, which serves as the topological invariant of the Weyl node. (d,e) A Dirac node arises from the crossing between two doubly degenerate bands (1, 2 and 3, 4). We can also enclose the Dirac node by a sphere in k space. The sphere will also satisfy the same two conditions. Because a Dirac node can be viewed as two degenerate Weyl nodes of opposite chirality, it can be shown that the chiral charge of a Dirac node is always zero, i.e., χ = 0. (g–i) The new band crossing here arises from the crossing between a singly degenerate band and a doubly degenerate band. However, if we try to enclose the triply degenerate node with a sphere, we see that it is not possible to have a fully gapped band structure on the sphere. Specifically, between bands 1 and 2, the band gap vanishes at the left-pole of the sphere. Similarly, between bands 2 and 3, the band gap is zero at the right-pole of the sphere. For this reason, it is not possible to define and calculate the Chern number on the sphere as done in the Dirac/Weyl cases. (f) A trivial case where the band crossing is a simple composition of a 0D Weyl node plus a 1D nodal line.