Vortices in Chern-Simons-Ginzburg-Landau Theory and the Fractional Quantum Hall Effect

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.