  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
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
Bernevig-Hughes-Zhang (BHZ) model. By implementing this algorithm, we propose a
novel 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 subtrates favor the HESs of quantum spin Hall system and which are not.
Correspondingly, our work demonstrates that the substrate related polaronic
effects have significant role on 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
non-topological ones.
