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

Enger, Håkon
Master thesis
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5163.pdf (1.188Mb)
Year
1998
Permanent link
http://urn.nb.no/URN:NBN:no-5073

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  • Fysisk institutt [1715]
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.
 
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