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We study electron transport at the edge of a generic disorderedtwo-dimensional topological insulator, where some channels are topologicallyprotected from backscattering. Assuming the total number of channels is large,we consider the edge as a quasi-one-dimensional quantum wire and describe it interms of a non-linear sigma model with a topological term. Neglectinglocalization effects, we calculate the average distribution function oftransmission probabilities as a function of the sample length. We mainly focuson the two experimentally relevant cases: a junction between two quantum Hall(QH) states with different filling factors (unitary class) and a relativelythick quantum well exhibiting quantum spin Hall (QSH) effect (symplecticclass). In a QH sample, the presence of topologically protected modes leads toa strong suppression of diffusion in the other channels already at the scalesmuch shorter than the localization length. On the semiclassical level, this isaccompanied by the formation of a gap in the spectrum of transmissionprobabilities close to unit transmission, thereby suppressing shot noise andconductance fluctuations. In the case of a QSH system, there is at most onetopologically protected edge channel leading to weaker transport effects. Inorder to describe `topological' suppression of nearly perfect transparencies,we develop an exact mapping of the semiclassical limit of the one-dimensionalsigma model onto a zero-dimensional sigma model of a different symmetry class,allowing us to identify the distribution of transmission probabilities with theaverage spectral density of a certain random-matrix ensemble. We extend ourresults to other symmetry classes with topologically protected edges in twodimensions.

Semiclassical electron transport at the edge of a 2D topological insulator: Interplay of protected and unprotected modes