Effective action of a compressible quantum Hall state edge: Application to tunneling

The electrodynamical response of the edge of a compressible quantum Hall system affects tunneling into the edge. Using composite Fermi liquid theory, we derive an effective action for the edge modes interacting with tunneling charge. This action generalizes the chiral Luttinger liquid theory of the quantum Hall edge to compressible systems in which transport is characterized by a finite Hall angle. In addition to the standard terms, the action contains a dissipative term. The tunneling exponent is calculated as a function of the filling fraction for several models, including screened and unscreened long-range Coulomb interaction, as well as a short-range interaction. We find that tunneling exponents are robust and to a large extent insensitive to the particular model. We discuss recent tunneling measurements in overgrown cleaved edge systems, and demonstrate that the profile of charge density near the edge is very sensitive to the parameters of the system. In general, the density is nonmonotonic, and can deviate from the bulk value by up to 30%. Implications for the correspondence to chiral Luttinger edge theories are discussed.