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

We study stability of multiple conducting edge states in a topological insulator against all multi-particle perturbations allowed by the time-reversal symmetry. We model a system as a multi-channel Luttinger liquid, where the number of channels equals the number of Kramers doublets at the edge. We show that in the clean system with N Kramers doublets there always exist relevant perturbations (either of superconducting or charge density wave character) which always open (N-1) gaps. In the charge density wave regime, (N-1) edge states get localised. The single remaining gapless mode describes sliding of 'Wigner crystal' like structure. Disorder introduces multi-particle backscattering processes. While the single-particle backscattering turns out to be irrelevant, the two-particle process may localise this gapless, in translation invariant system, 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 density-density repulsion parameters whereas any higher number N>2 of doublets gets fully localised by the disorder pinning irrespective of the parity issue.