<p>The fractional quantum Hall effect (FQHE), discovered in 1982 in a two-dimensional electron system, has generated a wealth of successful theory and new concepts in condensed matter physics, but is still not fully understood. The possibility of having nonabelian quasiparticle statistics has recently attracted attention on purely theoretical grounds but also because of its potential applications in topologically protected quantum computing.</p><p>This thesis focuses on the quasiparticles using three different approaches. The first is an effective Chern-Simons theory description, where the noncommutativity imposed on the classical space variables captures the incompressibility. We propose a construction of the quasielectron and illustrate how many-body quantum effects are emulated by a classical noncommutative theory.</p><p>The second approach involves a study of quantum Hall states on a torus where one of the periods is taken to be almost zero. Characteristic quantum Hall properties survive in this limit in which they become very simple to understand. We illustrate this by giving a simple counting argument for degeneracy 2<i>n</i>^-1, pertinent to nonabelian statistics, in the presence of 2<i>n</i> quasiholes in the Moore-Read state and generalise this result to 2<i>n</i>-<i>k</i> quasiholes and <i>k </i>quasielectrons.</p><p>In the third approach, we study the topological nature of the degeneracy 2<i>n</i>^-1 by using a recently proposed analogy between the Moore-Read state and the two-dimensional spin-polarized p-wave BCS state. We study a version of this problem where one can use techniques developed in the context of high-<i>T</i>c superconductors to turn the vortex background into an effective gauge field in a Dirac equation. Topological arguments in the form of index theory gives the degeneracy 2<i>n</i>^-1 for 2<i>n</i> vortices.</p>

The quantum Hall (QH) system---cold electrons in two dimensions in a perpendicular magnetic field---is a striking example of a system where unexpected phenomena emerge at low energies. The low-energy physics of this system is effectively one-dimensional due to the magnetic field. We identify an exactly solvable limit of this interacting many-body problem, and provide strong evidence that its solutions are adiabatically connected to the observed QH states in a similar manner as the free electron gas is related to real interacting fermions in a metal according to Landau's Fermi liquid theory. The solvable limit corresponds to the electron gas on a thin torus. Here the ground states are gapped periodic crystals and the fractionally charged excitations appear as domain walls between degenerate ground states. The fractal structure of the abelian Haldane-Halperin hierarchy is manifest for generic two-body interactions. By minimizing a local k+1-body interaction we obtain a representation of the non-abelian Read-Rezayi states, where the domain wall patterns encode the fusion rules of the underlying conformal field theory. We provide extensive analytical and numerical evidence that the Laughlin/Jain states are continuously connected to the exact solutions. For more general hierarchical states we exploit the intriguing connection to conformal field theory and construct wave functions that coincide with the exact ones in the solvable limit. If correct, this construction implies the adiabatic continuation of the pertinent states. We provide some numerical support for this scenario at the recently observed fraction 4/11. Non-QH phases are separated from the thin torus by a phase transition. At half-filling, this leads to a Luttinger liquid of neutral dipoles which provides an explicit microscopic example of how weakly interacting quasiparticles in a reduced (zero) magnetic field emerge at low energies. We argue that this is also smoothly connected to the bulk state.

The fractional quantum Hall effect (FQHE), discovered in 1982 in a two-dimensional electron system, has generated a wealth of successful theory and new concepts in condensed matter physics, but is still not fully understood. The possibility of having nonabelian quasiparticle statistics has recently attracted attention on purely theoretical grounds but also because of its potential applications in topologically protected quantum computing. This thesis focuses on the quasiparticles using three different approaches. The first is an effective Chern-Simons theory description, where the noncommutativity imposed on the classical space variables captures the incompressibility. We propose a construction of the quasielectron and illustrate how many-body quantum effects are emulated by a classical noncommutative theory. The second approach involves a study of quantum Hall states on a torus where one of the periods is taken to be almost zero. Characteristic quantum Hall properties survive in this limit in which they become very simple to understand. We illustrate this by giving a simple counting argument for degeneracy 2n-1, pertinent to nonabelian statistics, in the presence of 2n quasiholes in the Moore-Read state and generalise this result to 2n-k quasiholes and k quasielectrons. In the third approach, we study the topological nature of the degeneracy 2n-1 by using a recently proposed analogy between the Moore-Read state and the two-dimensional spin-polarized p-wave BCS state. We study a version of this problem where one can use techniques developed in the context of high-Tc superconductors to turn the vortex background into an effective gauge field in a Dirac equation. Topological arguments in the form of index theory gives the degeneracy 2n-1 for 2n vortices.