Stability of a topological insulator: Interactions, disorder, and parity of Kramers doublets

We study the stability of multiple conducting edge states in a topological insulator against all multiparticle perturbations allowed by time-reversal symmetry. We model a system as a multichannel Luttinger liquid, where the number of channels equals the number of Kramers doublets at the edge. We show that in a clean system with $N$ Kramers doublets there always exist relevant perturbations (either of a superconducting or charge density wave character) which always open $N\ensuremath{-}1$ gaps. In the charge density wave regime, $N\ensuremath{-}1$ edge states get localized. The single remaining gapless mode describes the sliding of a ``Wigner-crystal''-like structure. Disorder introduces multiparticle backscattering processes. While single-particle backscattering turns out to be irrelevant, the two-particle process may localize this gapless mode. Our main result is that an interacting system with $N$ Kramers doublets at the edge may be either a trivial insulator or a topological insulator for $N=1$ or 2, depending on the density-density repulsion parameters, whereas any higher number $N>2$ of doublets gets fully localized by disorder pinning, irrespective of the parity issue.