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By using the cluster perturbation theory, we investigate the effects of thelocal electron-phonon interaction in the quantum spin Hall topologicalinsulator described by the half-filled Kane-Mele model on an honeycomb lattice.Starting from the topological non trivial phase, where the minimal gap islocated at the two inequivalent Dirac points of the Graphene, $\text{K}$ and$\text{K}'$, we show that the coupling with quantum phonons induces atopological-trivial quantum phase transition through a gap closing andreopening in the $\text{M}$ point of the Brillouin zone. The average number offermions in this point turns out to be a direct indicator of the quantumtransition pointing out a strong hybridization between the two barequasiparticle bands of the Kane-Mele model. By increasing the strength ofcharge-lattice coupling, the phonon Green's propagator displays a two peakstructure: the one located at the lowest energy exhibits a softening that ismaximum around the topological transition. Numerical simulations provide alsoevidence of several kinks in the quasiparticle dispersion caused by thecoupling of the electrons with the bosonic lattice mode.

Topological phase transition in quantum spin Hall insulator in the presence of charge lattice coupling