Disorder induced phase transition in a Chern insulator as a percolation problem

Two-dimensional Chern insulators can undergo disorder-induced topological phase transitions, marked by the emergence of a critical delocalized state and the breakdown of edge modes. In this letter, we approach this problem from the perspective of impurity-bound states localized around vacancies, which we term \emph{ring states}. We derive a low-energy effective Hamiltonian describing the hybridization of these states and demonstrate that, for periodically arranged impurities, increasing impurity concentration can drive a topological phase transition. At the transition, the Chern number of the whole system changes through the threading of a $2\pi$ Berry flux, accompanied by the percolation of ring states. We further show that this picture extends to randomly distributed impurities, where the transition point can be identified by estimating the self-energy in the weak-disorder regime. Our analysis offers a new framework for understanding localization transitions in topological systems.