Transport in a one-dimensional Luttinger liquid

TL;DR

This paper reviews theoretical insights into transport phenomena in a one-dimensional Luttinger liquid, emphasizing impurity effects, bosonization techniques, and phenomena like fractional charge, with applications to quantum Hall edges and quantum dots.

Contribution

It provides a comprehensive review of transport in 1D Luttinger liquids, highlighting impurity effects, bosonization methods, and fractional quasiparticles, with applications to quantum Hall and quantum dot systems.

Findings

Rich and distinct transport behavior compared to non-interacting electrons

Existence of fractionally charged quasiparticles in the system

Analysis of impurity effects from different interaction regimes

Abstract

In this paper we review recent theoretical results for transport in a one-dimensional (1d) Luttinger liquid. For simplicity, we ignore electron spin, and focus exclusively on the case of a single-mode. Moreover, we consider only the effects of a single (or perhaps several) spatially localized impurities. Even with these restrictions, the predicted behavior is very rich, and strikingly different than for a 1d non-interacting electron gas. The method of bosonization is reviewed, with an emphasis on physical motivation, rather than mathematical rigor. Transport through a single impurity is reviewed from several different perspectives, as a pinned strongly interacting ``Wigner" crystal and in the limit of weak interactions. The existence of fractionally charged quasiparticles is also revealed. Inter-edge tunnelling in the quantum Hall effect, and charge fluctuations in a quantum dot under the conditions of Coulomb blockade are considered as examples of the developed techniques.