Fortunately, there is a way to work around this problem. The Chern numbers can only change when bands touch.[@Sim83a] Before and after a touching of bands, they differ by an integer. Only the Chern numbers of bands involved in the touching change. Furthermore, there is a sum-rule relating the Chern numbers before and after a band-touching. Suppose that, by varying a parameter in the Hamiltonian, one induces bands $n$ and $n+1$ to touch before moving apart again. Let ${\rm Ch}_m^<$ (${\rm Ch}_m^>$) be the Chern numbers before (after) the bands touched. Then ${\rm Ch}_n^<+{\rm Ch}_{n+1}^<={\rm Ch}_n^>+{\rm Ch}_{n+1}^>$, i.e. the sum of the two Chern numbers is preserved. If bands $n$ and $n+1$ are both occupied, $\sigma_{xy}$ stays constant although ${\rm Ch}_n$ and ${\rm Ch}_{n+1}$ change. Thus $\sigma_{xy}$ only changes if the Fermi energy lies in the gap between bands $n$ and $n+1$.

In order to calculate the change in Chern numbers, one only needs information about the bands that touch and only in regions of the Brillouin zone that are close to the points where the bands touch.[@Osh94; @Ono01] This information may be obtained by means of a perturbative expansion in the wave vector $\bm{k}$ and the external parameter that 