Transport in Single Channel Quantum Wires

This tutorial article gives an introduction to the methods needed to treat interacting electrons in a quantum wire with a single occupied band. Since one-dimensional Fermions cannot be described in terms of noninteracting quasiparticles, the Tomonaga-Luttinger model is presented in some detail with an emphasis on transport properties. To achieve a self-contained presentation, the Bosonization technique for one-dimensional Fermions is developed, accentuating features relevant for nonequilibrium systems. The screening of an impurity in the wire is discussed, and the insight gained on the electrostatics of a quantum wire is used to describe the coupling to Fermi-liquid reservoirs. These parts of the article should be readily accessible to students with a background in quantum mechanics including second quantization. To illustrate the usefulness of the methods presented, the current-voltage relation is determined exactly for a spin-polarized quantum wire with a particular value of the interaction parameter. This part requires familiarity with path integral techniques and connects with the current literature.