A non-perturbative expression for the transmission through a leaky chiral edge mode

TL;DR

This paper derives a non-perturbative, closed-form formula for the conductance of chiral edge modes in topological systems, accounting for impurities and bulk interactions, simplifying analysis of disordered quantum systems.

Contribution

It introduces a novel determinant ratio formula for the Green function of chiral edge modes coupled to disordered bulk, applicable to various topological systems.

Findings

Simplifies disorder averaging in 1D wires with impurities.

Allows estimation of phase shifts in quantum Hall systems with bulk impurities.

Provides a unified, non-perturbative approach to conductance in topological edge states.

Abstract

Chiral edge modes of topological insulators and Hall states exhibit non-trivial behavior of conductance in the presence of impurities or additional channels. We will present a simple formula for the conductance through a chiral edge mode coupled to a disordered bulk. For a given coupling matrix between the chiral mode and bulk modes, and a Green function matrix of bulk modes in real space, the renormalized Green function of the chiral mode is expressed in closed form as a ratio of determinants. We demonstrate the usage of the formula in two systems: i) a 1d wire with random onsite impurity potentials for which we found the disorder averaging is made simpler with the formula, and ii) a quantum Hall fluid with impurities in the bulk for which the phase picked up by the chiral mode due to the scattering with the impurities can be conveniently estimated.