Bosonic fractional Chern insulators from a strongly coupled fermionic Haldane-Holstein model

Recently, the interplay of topology and electron-electron correlation in topological flat band systems has received a lot of interest in condensed matter physics, particularly following the experimental observation of fractional quantum anomalous Hall effects. However, the role of electron-phonon coupling, which is another crucial correlation, is less explored in topological bands. In this work, we investigate this interplay of topology and electron-phonon coupling by studying the combined Haldane-Holstein model. In the strong-coupling, low-energy limit, the Haldane-Holstein model can be reduced to an effective electron-only model by integrating out the phonon degrees of freedom. We analyze this effective model in two distinct cases. For spinful electrons, strong electron-phonon coupling induces pairing between opposite spins, and a bosonic fractional Chern insulator (FCI) phase would emerge at a filling factor of ν=1/2 upon tuning the topological Haldane term. For spin-polarized electrons, the coupling generates longer-ranged, three-body hopping terms, which modify the original Haldane model and effectively lead to a more dispersive band. Using exact diagonalization, we further identify and map out a variety of emergent phases in the strongly coupled Haldane-Holstein model for both the spinful and spin-polarized cases.