Topological transitions as percolation of Berry curvature

We investigate the role of the sign structure of the Berry curvature in two-dimensional two-band Chern insulators. Motivated by subtleties that arise when attempting to construct Euler characteristics from the quantum metric in the presence of sign-changing Berry curvature, we focus on the connectivity properties of Berry-curvature sign domains in momentum space. For the Qi-Wu-Zhang model, the Haldane model, and an extended Haldane model with next-next-nearest-neighbor hopping, we decompose the Brillouin zone (BZ) into regions where the Berry curvature has the same sign as its global peak (peak-sign regions) and regions with the opposite sign. We show numerically that in topologically non-trivial phases with non-zero Chern number the oppositely signed regions form isolated, island-like clusters, whereas in topologically trivial phases they develop into percolating, river-like networks that span the BZ. This behavior is quantified by a percolation ratedefined as the fraction of-points belonging to the largest opposite-sign cluster, which exhibits a sharp crossover at phase boundaries between topologically trivial and non-trivial sectors in all three models. We further provide systematic tests of mesh-size dependence and noise robustness confirming that the percolation-based classification is stable under numerical and experimental uncertainties. Our results provide a simple geometric picture of how the system passes between topologically trivial and non-trivial phases in momentum space, and suggest a practical diagnostic that can be applied to experimentally reconstructed Berry-curvature maps.