Transport properties of a Luttinger liquid in the presence of several time-dependent impurities

TL;DR

This paper investigates how multiple time-dependent impurities affect electron transport in a Luttinger liquid, revealing interference effects and conditions for current enhancement, especially in the presence of weak repulsive interactions.

Contribution

It extends previous single-impurity studies to multiple impurities, uncovering interference phenomena and band-like structures in backscattering currents.

Findings

Current enhancement occurs only for strong repulsive interactions with a single impurity.

Interference between multiple impurities can induce positive backscattering current even for weak repulsive interactions.

Impurity frequency bands broaden with more impurities due to interference effects.

Abstract

We study the transport properties of a Luttinger liquid in the presence of several time-dependent weak point-like impurities. Our starting point is the bosonized form of the Luttinger liquid Hamiltonian with a potential introduced by the impurities. We find the correction to the total current due to the backscattering of the electrons by the impurities in first order of perturbation theory for a general time-dependent impurity potential. We then apply this result to a single time-dependent impurity, and show that it reproduces the result obtained by Feldman and Gefen in Ref. [7], in particular that the current enhancement occurs only for strong repulsive electron-electron interactions. For the case of two impurities oscillating with the same frequency, we find a novel effect, namely, that the dc-component of the backscattering current is positive {\it even for weak repulsive interactions}, due to the presence of the interference term induced by spatial correlations in the Luttinger liquid. We show that the impurity frequencies at which the enhancement takes place form a band-like structure. In the case of several impurities, the bands become broader, because of interference effects.