Sum of the Chern number of all bands

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}