A Microscopic Hierarchy Theory of the Fractional Quantum Hall Effect

Abstract

In this paper we present a microscopic hierarchy theory of the fractional quantum Hall effect. The wave functions of the ground state and the collective excitation states are obtained in terms of the electron coordinates. Working in the subspace spanned by the quasiparticles of the 1/m L Laughlin ground state, with m L an odd integer, it is shown that there exists a simple mapping between electron states in the quasiparticle subspace and states of an auxiliary boson system which is defined such that the number of the bosons is the same as that of the quasiparticles and the total magnetic flux quanta seen by the bosons equals the number of electrons. For the auxiliary boson system, one can write down the Laughlin state as well as the density wave states, analogous to the electron system at filling factor 1/m L . By mapping these states onto the quasiparticle subspace of the electrons, we find that the resulting wave functions provide a quite good description for the ground state and the collective excitations respectively of the original electron system at filling factor ν=1/(m L (± 1/2p)) with p a positive integer. This construction of the ground state and the collective excitation states can be repeated for higher filling factors. The theory presented in this paper can be viewed as a microscopic realization of Haldane's original hierarchy picture.