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Using density-matrix renormalization group simulations with local basis optimization, the authors examine how dynamical Holstein phonons destabilize integer and fractional Chern insulators into charge-ordered phases within a flat-band Haldane model. They show that electron-phonon coupling dresses neutral collective excitations into exciton and magnetoroton polarons whose mode softening directly anticipates the charge-ordering wave vectors. Crucially, the coupling imprints clear dispersive signatures on the phonon spectrum and induces a chirality deficit in the fractional Chern insulator's graviton response.
Phonon-driven destruction of fractional Chern insulators is mediated by the softening of hybrid magnetoroton polarons, exposing topological graviton modes and chirality deficits directly within measurable lattice phonon spectra.
Strong electron-phonon coupling can destabilize both Chern insulators (CIs) and fractional Chern insulators (FCIs) in favor of charge order, but whether the instabilities in CI and FCI share a common microscopic mechanism remains unclear. In this paper, we address this question by studying a flat-band Haldane model coupled to dynamical Holstein phonons at integer and fractional fillings using density-matrix renormalization group with local basis optimization. We find that the low-energy effects of phonons are dominated by their dressing of charge-neutral collective modes. In the CI and FCI, phonons dress excitons and magnetorotons into composite modes that we identify as exciton polarons and magnetoroton polarons, respectively. These modes soften strongly upon approaching the transition, while the momenta of their energy minima anticipate the ordering wave vectors of the charge-ordered phases. The phonon spectrum, in turn, acquires dispersive features inherited from the exciton and magnetoroton modes, providing direct lattice signatures of these neutral excitations. Within the FCI phase, electron-phonon coupling also enhances the opposite-chirality spectral weight of the graviton response, thereby resulting in a chirality deficit. Our results provide a unified excitation-based picture of phonon-driven instabilities in CIs and FCIs and establish lattice dynamics as a probe of their neutral collective modes.