Electron-phonon coupling induced topological phase transition in an α-T3 Haldane-Holstein model

We present impelling evidence of topological phase transitions induced by electron-phonon (e-ph) coupling in an alpha-T3 Haldane-Holstein model that facilitates smooth tunability between graphene (alpha = 0) and a dice lattice (alpha = 1). The e-ph coupling has been incorporated via the Lang-Firsov transformation which adequately captures the polaron physics in the high-frequency (anti-adiabatic) regime, and yields an effective Hamiltonian through zero phonon averaging at T = 0. While exploring the signature of phase transitions driven by polaron and its interplay with the parameter alpha, we identify two regions based on the values of alpha, namely, the low to intermediate range (0 < alpha <= 0.6) and larger values of alpha (0.6 < alpha < 1), where the topological transitions host distinct behavior. There exists a single critical e-ph coupling strength for the former, below which the system behaves as a topological insulator characterized by edge modes, finite Chern number, and Hall conductivity, with all of them vanishing above this value, and the system undergoes a spectral gap closing transition. Further, the critical coupling strength depends upon alpha. For the latter case (0.6 < alpha < 1), the scenario is more interesting where there are two critical values of the e-ph coupling at which trivial-topological-trivial and topological-topological-trivial phase transitions occur. Our study shows a significant difference with regard to the well-known unique transition occurring at alpha = 0.5 (or at 0.7) in the absence of the e-ph coupling, and thus underscores the importance of interaction effects on topological phase transitions.