# Substrate-limited helical edge states

## TL;DR

This paper analytically investigates how substrate-induced surface optical phonons affect the electronic and edge states of two-dimensional topological insulators, revealing their significant role in tuning quantum phase transitions.

## Contribution

It introduces a phonon-dressed BHZ Hamiltonian using Lee-Low-Pines variational approximation to study substrate effects on topological insulators and their edge states.

## Key findings

- Substrate-related polaronic effects significantly renormalize bulk and edge states.
- Certain substrates favor the emergence of helical edge states.
- Surface optical phonons can tune topological phase transitions.

## Abstract

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.