# Backscattering in a helical liquid induced by Rashba spin-orbit coupling and electron interactions: locality, symmetry, and cutoff aspects

## TL;DR

This paper investigates how backscattering in 1D helical liquids depends on the locality, symmetry, and spectrum cutoff, revealing conditions under which backscattering is suppressed or inevitable, thus clarifying previous conflicting results.

## Contribution

It provides a comprehensive analysis of backscattering effects considering general backscattering fields and interactions, resolving previous controversies and identifying key factors influencing backscattering.

## Key findings

- Backscattering is absent with local backscattering fields and symmetric, local interactions without spectrum cutoff.
- Violation of locality, symmetry, or presence of cutoff induces backscattering.
- Backscattering current exhibits distinct low-bias scaling regimes and a crossover between them.

## Abstract

The combination of the time-reversal-symmetric single-particle backscattering field (commonly known as Rashba spin-orbit coupling) and non-backscattering electron interactions is generally expected to produce inelastic backscattering in 1D helical electron liquids at the edge of 2D topological insulators, as theoretically predicted in a number of works. An opposite conclusion of absent backscattering was reached in a recent work [H.-Y. Xie et al., Phys. Rev. Lett. 116, 086603 (2016)] for the "local" model of the backscattering field and interactions. Motivated to resolve this potential controversy, in the present work, we study backscattering effects employing fermionic perturbation theory and considering quite general forms of the backscattering field and electron interactions. We discover that backscattering effects are crucially sensitive to the locality properties of the backscattering field and electron interactions, to the symmetry of the latter, as well as to the presence or absence of the cutoff of the electron spectrum. We find that backscattering is indeed absent under the following assumptions: (i) local backscattering field; (ii.a) local or (ii.b) SU(2)-symmetric interactions; (iii) absent cutoff of the edge-state spectrum. However, violation of any of these conditions leads to backscattering. This also reconciles with the results based on the bosonization technique. We calculate the associated backscattering current, establish its low-bias scaling behavior, and predict a crossover between two different scaling regimes. The main implication of our findings is that backscattering of some magnitude is inevitable in a real system, although could be quite suppressed for nearly local backscattering field and interactions.