  We first study the importance of the sign of the Berry curvature in the Euler
characteristic of the two-dimensional topological material with two bands. Then
we report an observation of a character of the topological transition as a
percolation of the sign of the Berry curvature. The Berry curvature F has peaks
at the Dirac points, enabling us to divide the Brillouin zone into two regions
depending on the sign of the F: one with the same sign with a peak and the
other with the opposite sign. We observed that when the Chern number is
non-zero, the oppositely signed regions are localized. In contrast, in the case
of a trivial topology, the oppositely signed regions are delocalized
dominantly. Therefore, the oppositely signed region will percolate under the
topological phase transition from non-trivial to trivial. We checked this for
several models including the Haldane model, the extended Haldane model, and the
QWZ model. Our observation may serve as a novel feature of the topological
phase transition.
