UBC Theses and Dissertations
Field theoretical quantities in the fractional quantum Hall effect Curnoe, Curnoe, Stephanie Hythe Stephanie Hythe
This thesis studies two models of the fractional quantum Hall effect (FQHE), the bosonic (Chern-Simons-Landau-Ginzburg) description and the fermionic (composite fermion gauge theory) description. The bosonic theory attempts to describe the FQHE states at filling fractions v = [formula] while the fermionic theory attempts to describe the states at v = [formula] and the metallic states in between. Within the bosonic theory, the fractionally charged quasiparticles of the FQH system are vortices which appear during the breakdown of the uniform quantum Hall state. The energetics of a single vortex state are studied whereby it is shown how the system may become unstable to the formation of vortices. Numerical vortex profiles are computed by minimising the Hamiltonian. Using the fermionic theory of composite fermions interacting with gauge fluctuations, we consider two important field theoretic quantities, the self-energy and the thermodynamic potential in a finite magnetic field. We find that the conventional Luttinger-Ward treatment of the oscillatory behaviour of the thermodynamic potential is not applicable in two dimensions, for any kind of interaction. Instead we propose a new formulation which omits all crossed graphs and which necessarily includes the oscillatory self-consistent self-energy. To second order in perturbation theory, the oscillatory self-energy is calculated by retaining Landau level quantisation on the internal fermion line. The low energy form of the self-consistent self-energy is obtained by means of a new iterative procedure which is introduced here. This procedure makes use of the structure introduced by Landau level quantisation. We also investigate the structure induced in the analogous two dimensional electron-phonon problem, in order to assist our understanding of the composite fermion self-energy. In the low energy limit, it is found that the renormalised form of the composite fermion Green's function is of the same form as the unrenormalised Green's function. Therefore we argue that the principal effects of interactions may be accounted for using a field-dependent renormalised mass. The iterative procedure for finding the self-consistent self-energy is used to evaluate the renormalised gap between the Fermi energy and the first excited states, which rapidly converges in a few iterations. We find a significant departure from the asymptotic result obtained by ignoring Landau level quantisation in the regime of experimentally relevant values of the parameters. We compare our findings with measurements of the gap in fractional Hall states near v = 1 / 2 .