The Fractional Quantum Hall Effect

Abstract

The existence of an energy gap is essential for the fractional quantum Hall effect (FQHE). However, in the case of the FQHE, the origin of the gap is different from that in the case of the IQHE. In the latter, the gap already exists in the single-electron spectrum. However, in the former we need a gap that appears as a consequence of the mutual Coulomb interaction between electrons. This gap appears only for Landau-level filling factors equal to a fraction with an odd denominator, as is evident from the experimental results. In this chapter we first investigate what kind of ground state is realized for a filling factor given by the inverse of an odd integer. Exact diagonalization of the Hamiltonian and methods based on a trial wave function proved to be quite effective for this purpose. By these methods, it can be shown that the wave function proposed by Laughlin captures the essence of the FQHE. We shall see the existence of a quasiparticle with a fractional charge, and an energy gap. Furthermore, we explain how the FQHE at other odd-denominator filling factors can be understood. Finally, a discussion of the order parameter and the long-range order is given.