Chern numbers on the Fermi surface of bcc iron

A metal whose Fermi surface contains sheets with nonzero Chern numbers is topologically nontrivial. This can occur when either spatial inversion ($P$) or time-reversal ($T$) symmetry is broken, and spin-orbit is present. Taking ferromagnetic iron as a prototypical $T$-broken metal, we determine the Chern indices of all the Fermi sheets, starting from a census of the isolated band touchings in the Brillouin zone. Although there are many band touching points carrying a topological charge, the Chern index vanishes for most Fermi sheets. The reason is that they surround $P$-invariant points in the BZ, so that the enclosed band-touching points come in pairs of equal and opposite charge. The exceptions are two small electron pockets on the [001] $\Gamma$H line parallel to the magnetization. Each of them encloses a single Weyl point, leading to Chern indices of $\pm 1$. The contribution of these two pockets to the anomalous Hall conductivity is given, modulo a ${\bf G}$-vector, by their reciprocal-space separation, as in a magnetic Weyl semimetal. In order to resolve the quantum of indeterminacy~${\bf G}$ we plot isocontours of the Berry phase calculated along [010] strings of $k$-points, which carry the same topological information as Fermi arcs in the (010) surface bandstructure.