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.