Chiral Luttinger liquids at the fractional quantum Hall edge

The edges of quantum Hall fluids behave as one-dimensional conductors. This article reviews electron transport into these edge states, covering both the theory based on the chiral Luttinger liquid and the experimental findings using electron tunneling as the probe. The first part of the review presents a basic description of this theory, including a derivation of the density of states, to provide a framework and language for discussing the experimental observations. The signature of the chiral Luttinger liquid is a power-law behavior for the density of states and the tunneling conductances. Experimentally, two techniques have been applied to study the tunneling conductance, using a gated point contact between two quantum Hall edges, or using a cleaved-edge barrier between an edge and a normal conductor. The point-contact method exhibits resonant tunneling, which appears to show some aspects of the Luttinger liquid, and the cleaved-edge method has yielded clear power-law dependences in the off-resonance conductances. Power-law behavior over many orders of magnitude is observed, confirming the Luttinger-liquid character of the edge states. However, the power-law exponents, while in agreement with finite-size numerical calculations, can differ from the universal values predicted by the Chern-Simon field theory. This disagreement is still not well understood. The review concludes with a brief survey of other one-dimensional conductors that have been studied to look for characteristics of the nonchiral Tomonaga-Luttinger liquid.