The combination of the time-reversal-symmetric single-particle backscattering field (commonly known as Rashba spin-orbit coupling) and nonbackscattering electron interactions is generally expected to produce inelastic backscattering in one-dimensional helical electron liquids at the edge of two-dimensional 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 it could be quite suppressed for nearly local backscattering field and interactions.

• Received 27 June 2017

DOI:https://doi.org/10.1103/PhysRevB.96.155134