Topological Protection from Random Rashba Spin-Orbit Backscattering: Ballistic Transport in a Helical Luttinger Liquid

The combination of Rashba spin-orbit coupling and potential disorder induces a random current operator for the edge states of a 2D topological insulator. We prove that charge transport through such an edge is ballistic at any temperature, with or without Luttinger liquid interactions. The solution exploits a mapping to a spin $1/2$ in a time-dependent field that preserves the projection along one randomly undulating component (integrable dynamics). Our result is exact and rules out random Rashba backscattering as a source of temperature-dependent transport, absent integrability-breaking terms.

Figure 1

$\delta$-type level spacing distribution $p(s)$ for the noninteracting helical edge model, obtained by numerical diagonalization of the Hamiltonian Eq. (3b) in momentum space. We sample every other level to account for the Kramers degeneracy. Here $s=|{ϵ}_n-{ϵ}_{n-2}|/\mathrm{\Delta }$ is the normalized level spacing near energy ${ϵ}_n$, while $\mathrm{\Delta }$ is the average of $|{ϵ}_n-{ϵ}_{n-2}|$ over the chosen set of levels. The parameter $\mathrm{\Xi }$ is the disorder correlation length in units of the inverse ultraviolet momentum cutoff [39, 49]. Inset: The broadening of the $\delta$-type distribution with different values of $\mathrm{\Xi }$ [39].

Figure 2

Typical single-particle delocalized wave functions for the helical model (a) and the Dyson model (b) at energy $ϵ=0$ (momentum space exact diagonalization [39]).