Substrate-limited helical edge states

We derived analytical results for the gapless edge states of two-dimensional topological insulators in the presence of electron-surface optical (SO) phonon interaction due to substrates. We followed an analytical algorithm, called the Lee-Low-Pines variational approximation in the conventional polaron theory, to examine the substrate-induced effects on both bulk and edge states of a two-dimensional topological insulator within the frame work of the Bernevig-Hughes-Zhang (BHZ) model. By implementing this algorithm, we propose a phonon-dressed BHZ Hamiltonian which allows one to investigate the effects of various substrates not only on bulk states but also on the associated gapless helical edge states (HESs). We found that both the bulk and HESs are significantly renormalized in the momentum space due to the substrate-related polaronic effects. The model we developed here clarifies which substrates favor the HESs of the quantum spin Hall system and which do not. Correspondingly, our work demonstrates that the substrate-related polaronic effects have a significant role in the emergence of HESs. In other words, we show that SO phonons due to substrates modify the electronic band topology of topological insulators together with the associated HESs, and therefore, they can be used to tune quantum phase transitions between topological insulators and nontopological ones.