Sum of the Chern number of all bands My understanding is that, if we sum the Chern numbers of all the bands in systems, they add up to zero. I believe this is a rigorous mathematical result. Is it possible to understand this physically? 
 A: The statement that the sum of all Chern numbers vanishes can be made even stronger. Namely, the total Berry curvature over all bands vanishes. The Berry curvature for a band $n$ is defined as
\begin{equation}
F_n(k_x,k_y) = i \left[ \left< \partial_{k_x} u_n \right | \partial_{k_y} u_n \rangle - \left< \partial_{k_y} u_n \right | \partial_{k_x} u_n \rangle \right].
\end{equation}
Inserting the identity operator $1=\sum_m \left| u_m \right> \left< u_m \right|$ (which holds at any momentum) yields
\begin{align}
F_n(k_x,k_y) & = i \sum_m \langle \partial_{k_x} u_n \left | u_m \right> \left< u_m \right| \partial_{k_y} u_n \rangle - i \langle \partial_{k_y} u_n \left| \partial_{k_x} u_n \right> \\
& = i \sum_m \langle \partial_{k_y} u_m \left | u_n \right> \left< u_n \right| \partial_{k_x} u_m \rangle - i \langle \partial_{k_y} u_n \left| \partial_{k_x} u_n \right>,
\end{align}
where in the last step we used
\begin{equation}
0 = \partial_{k_j} \left( \left< u_n \right| u_m \rangle \right) = \left< \partial_{k_j} u_n \right| u_m \rangle + \left< u_n \right| \partial_{k_j} u_m \rangle.
\end{equation}
Hence, the total Berry curvature summed over all bands vanishes:
\begin{align}
\sum_n F_n(k_x,k_y) & = i \sum_{m,n} \langle \partial_{k_y} u_m \left | u_n \right> \left< u_n \right| \partial_{k_x} u_m \rangle - i \sum_n \langle \partial_{k_y} u_n \left| \partial_{k_x} u_n \right> \\
& = i \sum_m \langle \partial_{k_y} u_m \left| \partial_{k_x} u_m \right> - i \sum_n \langle \partial_{k_y} u_n \left| \partial_{k_x} u_n \right> \\
& = 0,
\end{align}
since both sums run over all bands and where we used the completeness relation again in the second step. It follows that the sum of all Chern numbers vanishes:
\begin{equation}
\sum_n \mathcal C_n = \sum_n \int d^2k \, F_n(k_x,k_y) = \int d^2k \sum_n  F_n(k_x,k_y) = 0.
\end{equation}
