Geometric Aspects and Neutral Excitations of the Fractional Quantum Hall Effect

Abstract

In this thesis, I will present studies on the collective modes of the fractional quantum Hall states, which are bulk neutral excitations reflecting the incompressibility that defines the topological nature of these states. It was first pointed out by Haldane that the non-commutative geometry of the fractional quantum Hall effects (FQHE) plays an important role in the intra-Landau-level dynamics. The geometrical aspects of the FQHE will be illustrated by calculating the linear response to a spatially varying electromagnetic field, and by a numerical scheme for constructing model wavefunctions for the neutral bulk excitations. Compared to early studies of the magneto-roton modes with single mode approximation (SMA), the scheme presented in this thesis is good not only in the long wavelength limit, but also for large momenta where the neutral excitations evolve into quasihole-quasiparticle pair. It is also shown that in the long wavelength limit, the SMA scheme produces exact model wavefunctions describing a quadrupole excitation. The same scheme can also extend to describe the neutral fermion mode in the Moore-Read state, reflecting its non-Abelian nature.

The numerically generated model wavefunctions are then identified with a family of analytic wavefunctions that describe both the magneto-roton modes and the neutral fermion modes. Like the ground state wavefunction of the Laughlin and Moore-Read state, the family of the analytic wavefunctions do not have any variational parameters. This set of analytic wavefunctions unifies previous numerical works on neutral excitations of single-component FQH states, both from the Jack polynomial point of view presented in this thesis, and from the composite fermion picture developed by Jain and collaborators. The compact analytic forms also lend much insight into the nature of the neutral excitations from the plasma analogy. In particular, the quadrupole excitation gap is related to the free energy cost of the fusion of charged particles in a two-dimensional plasma with a neutralizing background.