A conformal field theory description of fractional quantum Hall states

In this thesis, we give a description of fractional quantum Hall states in terms of conformal field theory (CFT). As was known for a long time, the Laughlin states could be written in terms of correlators of chiral vertex operators of a c=1 CFT. It was shown by G. Moore and N. Read that more general constructions are possible, and they proposed a new quantum Hall state, which has an underlying pairing structure. Related to that, it turned out that the quasiparticle excitations obey non-abelian statistics. The machienery of conformal field theory can be used to define and study more general paired, or clustered quantum Hall states, with or without spin. In this thesis, we focused on two classed of clustered (non-abelian) quantum Hall states, namely a spin polarized series, defined by N. Read and E. Rezayi and a spin-singlet series, which we proposed. These states, with or without quasiparticles, can be defined as conformal correlators in certain CFTs with an underlying affine Lie algebra symmetry. We determined the K-matrices, which describe the topological properties of the quantum Hall states. To find them, we used a generalization of the concept, to include composite particles, and, related to that, so called pseudoparticles, which account for the non-abelian statistics. In fact, clustering structure of the states is intimately related to the non-abelian statistics of the quasiparticles. The K-matrices also play an important role in the characters of the conformal field theories. Using the methods alluded to above, we studied the properties of the quasiparticle excitations. Another way of studying these quasiparticle exciations is by numerically diagonalizing electrons on the sphere, in the presence of a magnetic field. The interactions between the electrons are taken in such a way that the quantum Hall state is the unique ground state for a certain amount of flux quanta. By increasing the flux, quasiparticles are induced, and the groud state becomes degenerate. This degenaracy can be understood in terms of the conformal field theory describing the quantum Hall states and we were able to give an explicit formula for this degenaracy.