Anyons and the Fractional Quantum Hall Effect

Abstract

The only known physical objects which can be described as anyons are the quasi-particle and quasi-hole excitations of planar systems of electrons exhibiting the fractional quantum Hall effect (QHE) (for a review see for instance (Prange and Girvin 1990)). Actually most of the great interest that anyonic theories have attracted in the past few years derives precisely from their relevance to a better understanding of the fractional QHE (Halperin 1984), in conjunction with several claims that anyons can provide also a non-standard explanation of the mechanism of high temperature superconductivity (Chen et al. 1989). It is far beyond the scope of these lecture notes to treat these issues in a systematic and adequate way, and therefore we refer the reader to the many good reviews already existing in the literature, for example (Wen and Zee 1989b; Arovas 1989; Lykken et al. 1991). However, for the sake of completeness we think necessary to spend some time on at least one of these physical applications in order to convey the idea that anyons are not just mathematical fantasies. Since recent experiments have cast some shadow on the relevance of fractional statistics to the observed high temperature superconductivity (Lyons et al. 1990; Kiefl et al. 1990; Spielman et al. 1990), here we will concentrate only on the application of anyons to the theory of the fractional QHE. Of course our presentation will be schematic and not at all exhaustive. Furthermore, we will concentrate more on the formal aspects than on the condensed matter issues.