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

We study electron transport at the edge of a generic disordered two-dimensional topological insulator, where some channels are topologically protected from backscattering. Assuming the total number of channels is large, we consider the edge as a quasi-one-dimensional quantum wire and describe it in terms of a non-linear sigma model with a topological term. Neglecting localization effects, we calculate the average distribution function of transmission probabilities as a function of the sample length. We mainly focus on the two experimentally relevant cases: a junction between two quantum Hall (QH) states with different filling factors (unitary class) and a relatively thick quantum well exhibiting quantum spin Hall (QSH) effect (symplectic class). In a QH sample, the presence of topologically protected modes leads to a strong suppression of diffusion in the other channels already at the scales much shorter than the localization length. On the semiclassical level, this is accompanied by the formation of a gap in the spectrum of transmission probabilities close to unit transmission, thereby suppressing shot noise and conductance fluctuations. In the case of a QSH system, there is at most one topologically protected edge channel leading to weaker transport effects. In order to describe `topological' suppression of nearly perfect transparencies, we develop an exact mapping of the semiclassical limit of the one-dimensional sigma model onto a zero-dimensional sigma model of a different symmetry class, allowing us to identify the distribution of transmission probabilities with the average spectral density of a certain random-matrix ensemble. We extend our results to other symmetry classes with topologically protected edges in two dimensions.