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

TL;DR

This paper investigates how topologically protected and unprotected edge modes influence electron transport in disordered 2D topological insulators, revealing suppression of diffusion and shot noise due to topological effects.

Contribution

It introduces a sigma model approach to analyze transmission probabilities in topological insulator edges, highlighting the impact of protected modes on transport properties.

Findings

Protected modes suppress diffusion in unprotected channels.

Formation of a gap near perfect transmission reduces shot noise.

Weak effects in quantum spin Hall systems with fewer protected channels.

Abstract

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.