The standard method for numerically calculating the Berry curvature of a 2D condensed matter system is given by Fukui-Hatsugai-Suzuki in this paper. They discretize $k$-space into a grid with tiny rectangles and calculate the Berry curvature on each rectangle using so-called overlap matrices $U_\mu$ and difference methods for derivatives. However, their definition of $U_\mu$ involves only eigenstates of the band in question (as in $U_\mu=\langle n(k) | n(k+\mu)\rangle$).
However, the general definition of Berry curvature (as from this popular reference) explicitly involves >1 bands in the expression (unlike $U_\mu$). Berry curvature $\Omega$ is also known as an interband quantity. So, my question is, why is it okay to still use an intraband $U_\mu$ to calculate $\Omega$ in multiband systems? How does $U_\mu$ incorporate effects of other bands? Or, have I misunderstood the application of the numerical scheme in the first reference? I know that the multiband formula for $\Omega$ comes from inserting the identity $\Sigma_p |p\rangle\langle p|=1$ to $\Omega=\nabla \times A$ (the berry connection, which is intraband like $U_\mu$).
I know that for 2-band systems, $\Omega_m$ = $-\Omega_n$ for energy bands $m,n$. However, will $U_\mu$ be appropriate even when this symmetry is not present, as in >2-band systems? So, I wanted how $U_\mu$ incorporates other bands' effects better, and confirm that it is valid for use in multiband systems (as the authors claim), but with the intraband $U_\mu$ I gave above. Thanks.