Chern-Simons-Ginzburg-Landau (CSGL) theory is an attempt of aphenomenological description of the fractional quantum Hall effect.The CSGL theory is studied mainly without considering the directapplications of the results. Vortices in CSGL theory are believed tobe the analogue of quasiparticles in the fractional quantum Halleffect. The details of the vortices are studied both analytically andnumerically, and we compare the analytical results to the numericalones. We show how the vortices may be understood as particles inMaxwell-Chern-Simons (MCS) theory. We solve the CSGL equations for avortex numerically for a range of the dimensionless parameter, andshow how the size and energy of a vortex depends on this parameter.We also study the connection between the CSGL theory and the GL andMCS theories numerically, and find support for our analytical results.
Also studied are various extensions of the CSGL theory. Theseextensions are made by adding terms to the CSGL Lagrangian. Theextended theories are mainly studied numerically. The first extensionwe study is the addition of a dynamical magnetic field. We show howthe charge is no longer quantized when the magnetic field is madedynamical. We also show how the inclusion of a dynamical magneticfield changes the size, energy and charge of a vortex, and we findthat the self-dual point of pure CSGL theory extends to a self-dualline.
The second extension we study is the extension of the CSGL wavefunction to a two-component spinor. We show how this extension allowsanother kind of vortex solutions, known as skyrmions, and show how thesize and spin of the skyrmions depend on the effective gyromagneticratio, and we reproduce qualitative results found by a different kindof study of a spin-dependent model for the fractional quantum Halleffect. Using our numerical results, we obtain a phase diagram forthe spin dependent CSGL theory.
The last part of the thesis is devoted to the duality between the CSGLtheory and the MCS theory. We make a detailed derivation of theduality starting from the Lagrangian of CSGL theory. We attempt touse this duality to find a better description of the dynamics ofvortices and a dispersion relation for a system with a gas of freevortices. We conclude that in this area there is still room forfurther study.