Abstract
Chern-Simons-Ginzburg-Landau (CSGL) theory is an attempt of a
phenomenological description of the fractional quantum Hall effect.
The CSGL theory is studied mainly without considering the direct
applications of the results. Vortices in CSGL theory are believed to
be the analogue of quasiparticles in the fractional quantum Hall
effect. The details of the vortices are studied both analytically and
numerically, and we compare the analytical results to the numerical
ones. We show how the vortices may be understood as particles in
Maxwell-Chern-Simons (MCS) theory. We solve the CSGL equations for a
vortex numerically for a range of the dimensionless parameter, and
show how the size and energy of a vortex depends on this parameter.
We also study the connection between the CSGL theory and the GL and
MCS theories numerically, and find support for our analytical results.
Also studied are various extensions of the CSGL theory. These
extensions are made by adding terms to the CSGL Lagrangian. The
extended theories are mainly studied numerically. The first extension
we study is the addition of a dynamical magnetic field. We show how
the charge is no longer quantized when the magnetic field is made
dynamical. We also show how the inclusion of a dynamical magnetic
field changes the size, energy and charge of a vortex, and we find
that the self-dual point of pure CSGL theory extends to a self-dual
line.
The second extension we study is the extension of the CSGL wave
function to a two-component spinor. We show how this extension allows
another kind of vortex solutions, known as skyrmions, and show how the
size and spin of the skyrmions depend on the effective gyromagnetic
ratio, and we reproduce qualitative results found by a different kind
of study of a spin-dependent model for the fractional quantum Hall
effect. Using our numerical results, we obtain a phase diagram for
the spin dependent CSGL theory.
The last part of the thesis is devoted to the duality between the CSGL
theory and the MCS theory. We make a detailed derivation of the
duality starting from the Lagrangian of CSGL theory. We attempt to
use this duality to find a better description of the dynamics of
vortices and a dispersion relation for a system with a gas of free
vortices. We conclude that in this area there is still room for
further study.