Interaction of Composite Fermions with a Gauge Field in the Fractional Quantum Hall Regime.

This thesis studies the problem of two-dimensional fermions interacting with a gauge field. This problem arises in a theory of the half-filled Landau level in connection with the composite fermion theory of the fractional quantum Hall effect. A composite fermion is generated by attaching even number of flux quanta to an electron. The transformation from the electron to the composite fermion can be realized by introducing an appropriate Chern-Simons gauge field. Especially, at the filling fraction nu = 1/2, composite fermions see effectively zero magnetic field at the mean field level because of the cancellation between the average of the Chern-Simons gauge field (from the attached magnetic flux quanta) and the external magnetic field. Thus, at the mean field level the system can be described as a Fermi liquid of composite fermions. In this thesis, the effect of the gauge-field fluctuations around the mean-field Fermi-liquid state has been studied. It turns out that singular behavior appears in the lowest-order self-energy correction of fermions by the transverse gauge-field fluctuation. This singular -self energy correction makes the effective mass of the fermion divergent so that the usual single particle picture breaks down. However, the one-particle Green's function is not gauge-invariant, so the singular self-energy could be an artifact of the gauge choice. This consideration leads us to examine the lowest order perturbative corrections to the gauge-invariant density-density and the current -current correlation functions. It is found that there are important cancellations between the self-energy corrections and the vertex corrections due to the Ward-identity. As a result, the density-density and the current-current correlation functions show a Fermi-liquid behavior for all ratios of omega and v_{F }q. From these results, one may suspect whether the divergent mass obtained from the self-energy has any physical meaning. In order to answer the question about the effective mass, it is important to examine other gauge-invariant quantities which may potentially show a divergent effective mass. We have calculated the corrections to the activation energy gap and the corresponding effective mass by looking at the compressibility of the system. They are turned out to be singular and consistent with the previous self -consistent treatment of the self-energy. Therefore, the divergent effective mass does have a physical meaning. At this point, it is clear that we need a unified framework to understand the apparently different behaviors of these two different results. In order to achieve this, we construct a quantum Boltzman equation (QBE) which describes all the low energy physics of the composite fermion system, and provide consistent explanation for the previous calculations of response functions. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617 -253-5668; Fax 617-253-1690.).