This is remarkable since the magnetic field through the sample averages to zero.

Here I briefly review the general theory for this phenomenon, which is known as the anomalous Hall effect.[@Kar54; @Thou82; @Thou85; @Sim83a; @Sim83b; @Koh84; @Osh94; @Ono01] It states that the Hall conductivity $\sigma_{xy}$ may be non-zero, even in the absence of a net magnetic flux through the sample, provided time-reversal invariance is still broken. Furthermore it is quantized in units of $e^2/h$ and can be written in terms of a set of integer topological invariants as $\sigma_{xy}=(e^2/h)\sum_n {\rm Ch}_n$.[@Thou82; @Thou85; @Sim83a; @Sim83b; @Koh84; @Osh94; @Ono01] Here $n$ is the discrete index that labels the energy bands and the sum ranges over all occupied bands. (The theory assumes that the Fermi energy is in a gap between bands so that there are no partially filled bands.) The invariant ${\rm Ch}_n$ is known as a Chern number, and can be calculated if the Bloch state for band $n$ is known throughout the whole Brillouin zone. Since only the Bloch states in the vicinity of $\bm{k}=0$ and only for the two bands that touch at $\varepsilon=0$ were found in Sec. \[near\_zero\], direct calculation of the ${\rm Ch}_n$ is not feasible here.

