| Graphene has emerged as an important two dimensional electron system with novel physical properties due to its relativistic-like linear energy-momentum dispersion relation at low energy. Alongside two dimensional electron systems in semiconductor heterostructures, it has a rich set of integer and fractional quantum Hall states. Significant progresses have been made recently, but a full understanding of these states is still lacking. The prevailing approach for fractional quantum Hall effects in graphene has been the numerical exact diagonalization. In this work, we develop a fermionic Chern-Simons effective theory for Dirac fermions as a complement to the existing theories, and to bring new insights in our understanding of the phenomena. In particular, we study the possibility for quantum Hall plateaus at even-denominator filling factors.;We first construct a unitary Chern-Simons transformation to attach even number of flux quanta to Dirac fermions. To deal with the four-fold spin-valley degeneracy, a set of K-matrices is introduced. At even-denominator filling factors in the zeroth Landau level, the fictitious magnetic field of the Chern-Simons field cancels the external magnetic field on average. It is shown that the Chern-Simons field mediates an effective mutual statistical interaction between composite Dirac fermions.;We further show the statistical interaction and Coulomb interaction favor the formation of an exciton condensate. Quasi-particles at finite filling factors can be regarded as excitations above the exciton condensate, and can be described as massive Dirac fermions. This means a mass is generated dynamically for Dirac fermions. Different types of K-matrices give rise to different mass gaps. The Chern numbers associated with different massive Dirac band structures can be used to classify the K-matrices. With different K, we can describe states at filling factor v = v˜, ± (−1 + v˜ ), ± (−2 + v˜) with v˜ < 1.;In the last part of the thesis, we study the pairing instability of the composite Dirac fermion liquid. We show the statistical interaction drives a complex p-wave pairing among the quasi-particles. As long as the Coulomb pair breaking effect is weak, the system can develop a superconducting energy gap, thus form a fractional quantum Hall state. |
