Studies of Composite Fermions in Fractional Quantum Hall Effect

Various aspects of the composite fermion theory are discussed in this thesis. In chapter one we give an introduction on what the fractional quantum Hall effect (FQHE) is and what the composite fermion theory of the FQHE tells us. Then we proceed to discuss in details the projection of composite fermion wave functions onto the lowest Landau Level (LL) in chapter two. In the large magnetic field limit, the excitation spectrum of free composite fermions are given and match well with those of the corresponding interacting electron systems. But the one-to-one correspondence between states of the non-interacting composite fermion systems and those of the corresponding non-interacting electron systems breaks down beyond the lowest band. In other words, non-interacting composite fermions behave differently from non-interacting electrons when they are excited to higher bands. We also show that the collective mode branches of incompressible states are well described as the collective modes of composite fermions. Our results suggest that, at small wave vectors, there is a single well-defined collective mode for all FQHE states. These are the issues which we will address in chapter three. In chapter four, we move on to discuss a theory of the FQHE in the limit of vanishing Zeeman energy. Motivated by the experimental observation of spin-unpolarized ground states at filling factors {2over 3}, {3over 5}, and {4 over 3}, we developed a microscopic theory for the FQHE in the limit of vanishing Zeeman energy, within the framework of the composite fermion theory. Our theory gives an explanation about the existence of spin-unpolarized ground states at these filling factors, and predicts the spin quantum numbers of the ground states, which are in agreement with numerical calculations. Furthermore, the trial wave functions of this theory are found to be extremely close to the actual Coulomb ground states. We also investigated the spectrum of interacting electrons at arbitrary filling factors in the limit of vanishing Zeeman splitting, and found that our theory successfully explains the low-energy spectrum, provided that the composite fermions interact with each other via a hard core potential. The electron system on a sphere with a magnetic monopole in the center resembles an atom. We investigated the applicability of the Hund's rule to this "FQHE atom" in the vanishing Zeeman energy limit in chapter five. We found that an application of the Hund's rule to electrons always predicts a wrong spin for the ground state, but the application of the Hund's rule to composite fermions generally predicts the ground state spin correctly. Thus, this atom should be viewed as an atom of composite fermions.