Universality in the Fractional Quantum Hall effect

In this lectures we review the fermion field theoretic approach to the Fractional Quantum Hall Effect and use it to discuss the origin of its remarkable universality. We discuss the semiclassical expansion around the average field approximation (AFA). We reexamine the AFA and the role of fluctuations. We argue that, order-by-order in the semiclassical expansion, the response functions obey the correct symmetry properties required by Galilean and Gauge Invariance and by the incompressibility of the fluid. In particular, we find that the low-momentum limit of the semiclassical approximation to the response functions is exact and that it saturates the f-sum rule. We discuss the nature of the spectrum of collective excitations of FQHE systems in the low-momentum limit. We applied these results to the problem of the screening of external charges and fluxes by the electron fluid, and obtained asymptotic expressions of the charge and current density profiles, for different types of interactions. The universality of the FQHE is demonstrated by deriving the form of the wave function of the ground state at long distances. We show that the wave functions of the fluid ground states of Fractional Quantum Hall systems, in the thermodynamic limit, are universal at long distances and that they have a generalized Laughlin form. This universality is a consequence of the analytic properties of the equal-time density correlation functions at long distances.