Weyl semimetals and monopoles
Weyl nodes act as sources or sinks of Berry curvature in three-dimensional momentum space.
$$\frac{1}{2\pi}\oint_{S^2}\boldsymbol\Omega\cdot d\mathbf S=\chi\in\mathbb Z$$
What this means
Weyl nodes act as sources or sinks of Berry curvature in three-dimensional momentum space.
Deep dive: mathematical and physical mechanism
Near a simple Weyl point, $H\approx\sum_i v_i q_i\sigma_i$. The Berry flux through a small sphere gives the chirality $\chi=\mathrm{sgn}(\det v)$ up to band convention. Two-dimensional slices between nodes can carry nonzero Chern number, yielding surface Fermi arcs.
How it appears in quantum phenomena
Weyl semimetals show Fermi arcs and anomalous transport.
Assumptions, caveats and limitations
Weyl nodes occur in gapless systems; protection differs from gapped-insulator topology.
Connect the ideas
For context, see the Topological Band Theory topic guide and the reading references.