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 $\ensuremath{\alpha}\text{\ensuremath{-}}{T}_{3}$ Haldane-Holstein model that facilitates smooth tunability between graphene ($\ensuremath{\alpha}=0$) and a dice lattice $(\ensuremath{\alpha}=1)$. 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 $\ensuremath{\alpha}$, we identify two regions based on the values of $\ensuremath{\alpha}$, namely, the low to intermediate range $(0<\ensuremath{\alpha}\ensuremath{\le}0.6)$ and larger values of $\ensuremath{\alpha}\phantom{\rule{0.28em}{0ex}}(0.6<\ensuremath{\alpha}<1)$, 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 $\ensuremath{\alpha}$. For the latter case $(0.6<\ensuremath{\alpha}<1)$, 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 $\ensuremath{\alpha}=0.5$ (or at 0.7) in the absence of the e-ph coupling, and thus underscores the importance of interaction effects on topological phase transitions.