Universality in the Fractional Quantum Hall Effect

This paper reviews the fermion field theoretic approach to the Fractional Quantum Hall Effect (FQHE) and discusses the origin of its remarkable universality. The methodology employed includes a semiclassical expansion around the average field approximation (AFA), with a reexamination of the AFA and the role of fluctuations. The authors argue that, order-by-order in the semiclassical expansion, the response functions obey the necessary symmetry properties mandated by Galilean and Gauge Invariance and the incompressibility of the fluid. Notably, it is demonstrated that the low-momentum limit of the semiclassical approximation to the response functions is exact and saturates the f-sum rule. The research delves into the nature of the spectrum of collective excitations of FQHE systems in the low-momentum limit and applies these results to the screening of external charges and fluxes by the electron fluid. Asymptotic expressions for charge and current density profiles are derived for various interaction types. The universality of the FQHE is established by deriving the wave function of the ground state at long distances, showing that the fluid ground states exhibit universal behavior at long distances with a generalized Laughlin form. This universality arises from the analytic properties of the equal-time density correlation functions at long distances.