Electron-phonon coupling induced topological phase transition in an \(\alpha\)-\(T_{3}\) Haldane-Holstein model

Abstract

We present impelling evidence of topological phase transitions induced by electron-phonon (e-ph) coupling in an \(\alpha\)-\(T_3\) Haldane-Holstein model that presents 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 of the system through zero phonon averaging at \(T=0\). While exploring the signature of the phase transition 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 \((0 < \alpha \le 0.6)\) and larger values of \(\alpha~(0.6 < \alpha < 1)\) where the topological transitions show distinct behaviour. 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 for \(\alpha\) in the range \([0.6:1]\). Our studies on e-ph coupling induced phase transitions show 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 underscore the importance of interaction effects on the topological phase transitions.