Nuclear Physics B (Proc. Suppl.) 33C (1993) 67-91 North-Holland I |[l[lll tr,-I "tl ",,il 5"ii,,,"1[15,"!1 "] PROCEEDINGS SUPPLEMENTS Universality in the Fractional Quantum Hall Effect Eduardo Fradkin ~ and Ana Lopez ~ ~Department of Physics, University of Illinois at Urbana-Chalnpaign 1110 W. Green St.,Urbana, IL61801, USA 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 semiclmssical expansion, the response functions obey the correct symmetry properties reqnired 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 linfit. 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 univex-sality of the FQHE is demonstrated by deriving the form of the wave ftmction 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 therxnodynamic 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. 1. Introduction Ever since its discovery in 1982, the Frac- tional Quantum Hall Effect (FQHE) has fasci- nated both experimentalists and theorists alike. The most salient feature of this remarkable effect is the very existence of the effect itself which has been checked with remarkable degree of precis- sion. The fact that the Hall plateaus are seen so clearly in materials which have quite complex lnicroscopic properties points to the universality of the phenomenon. It was recognized very early on that while some degree of disorder is essential to make the FQHE observable, the precission to which the effect is observed is due to the topologi- cal invariance of the Hall conductance in systems with an energy gap. Various theoretical approaches have been pro- posed to explain the FQHE. The Laughlin- Haldalm-Halperill [1-3] approach is based on the Laughlil~ variational ansatz for the ground state wave function. The Laughlin wave function gives the correct value for the Hall conductance, and yields an excellent ground state energy [1, 2]. Later on, Halperin [3] realized that the quasi- particles supported by this state exhibit not only fractional charge but that they are anyons, par- ticles with fi'actional statistics [4]. A hierar- chy of daughter states at other fractions different from the fundamental fractions, i.e., u = 1 can 777 ~ be constructed by considering a Laughlin-type ground state of the fractionally charged quasi- particles defined relative to the parent state one step up in the hierarchy. The higher-order FQHE states occur at a sequence of rational filling fi'ac- tions. Related to this approach is the composite fermion theory of the FQHE developed by Jain [5]. He found that the low energy states of the FQHE can be described in terms of weakly in- teracting composite fermions, where a composite ferlnion is an electron bound to an even number of vortices. He also proposed simple Jastrow-Slater trial wavefunctions for the incompressible FQHE states as well as for their low energy excitations. The validity of these wavefnnctions was confirlned by calculating numerically their overlap with the true Coulolnb states for systems with small num- ber of particles. Another approach consists on an effective Landau-Ginzburg field theory for the FQHE [6]. It was shown that the mean field solutions and the small fluctuations of the Landau-Ginzburg effec- tive action give a correct qualitative description of the physics of the low-energy degrees of free- 0920-5632/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved.