Real-Space Topology and Charge Order in the Haldane-Holstein Model.

We study the half-filled Haldane-Holstein model, where a paradigmatic Chern insulator is coupled to fully dynamical phonons, and provide an unbiased characterization of how retarded electron-phonon interactions destabilize Chern topology. Using determinant quantum Monte Carlo, we find that increasing the coupling drives an abrupt, first-order transition from a Chern insulator to a staggered charge-density wave that acts as a dynamical sublattice (Semenoff) mass. The transition is simultaneously signaled by a nearly quantized many-body Bott index and a real-space local Chern marker constructed from the interacting Green's function, both of which collapse as the charge order parameter becomes extensive. Spectral and open-boundary calculations reveal concomitant gap closing and the loss of boundary spectral weight at the critical coupling. Despite the generic phase problem induced by broken time-reversal symmetry, we show that it remains mild in the low-frequency regime and that the average phase factor sharply tracks the Chern insulator-charge density wave boundary. Our results establish a concrete route by which electron-phonon coupling can trigger a discontinuous collapse of Chern topology and provide experimentally relevant signatures for correlated topological platforms.