The eigenstates of interacting electrons in the fractional quantum Hall phase typically form fairly well-defined bands in the energy space. We study systems of six and eight electrons for filling factor 3/7>\ensuremath{\nu}> 2) / 7 and show that composite fermion theory gives insight into the origin of these bands and provides an accurate and complete microscopic description of the strongly correlated many-body states in the low-energy bands. This implies that, somewhat like in Landau's Fermi liquid theory, there is a one-to-one analogy between the low-energy Hilbert space of strongly interacting electrons in the fractional quantum Hall effect and that of weakly interacting electrons in the integer quantum Hall effect.
The eigenstates of interacting electrons in the fractional quantum Hall phase typically form fairly well-defined bands in the energy space. We study systems of six and eight electrons for filling factor 3/7>\ensuremath{\nu}> 2) / 7 and show that composite fermion theory gives insight into the origin of these bands and provides an accurate and complete microscopic description of the strongly correlated many-body states in the low-energy bands. This implies that, somewhat like in Landau's Fermi liquid theory, there is a one-to-one analogy between the low-energy Hilbert space of strongly interacting electrons in the fractional quantum Hall effect and that of weakly interacting electrons in the integer quantum Hall effect.
Exact diagonalization studies have revealed that the energy spectrum of interacting electrons in the lowest Landau level splits, nonperturbatively, into bands, which is responsible for the fascinating phenomenology of this system. The theory of nearly free composite fermions has been shown to be valid for the lowest band, and thus to capture the low-temperature physics, but it overpredicts the number of states for the excited bands. We explain the state counting of higher bands in terms of composite fermions with an infinitely strong short-range interaction between an excited composite-fermion particle and the hole it leaves behind. This interaction, the form of which we derive from the microscopic composite-fermion theory, eliminates configurations containing certain tightly bound composite-fermion excitons. With this modification, the composite-fermion theory reproduces, for all well defined excited bands seen in exact diagonalization studies, an exact counting for $\ensuremath{\nu}>1/3$, and an almost exact counting for $\ensuremath{\nu}\ensuremath{\le}1/3$. The resulting insight clarifies that the corrections to the nearly free composite-fermion theory are not thermodynamically significant at sufficiently low temperatures, thus providing a microscopic explanation for why it has proved successful for the analysis of the various properties of the composite-fermion Fermi sea.
Single particle basis functions for composite fermions are obtained from which many-composite fermion states confined to the lowest electronic Landau level can be constructed in the standard manner, i.e. by building Slater determinants. This representation enables a Monte Carlo study of systems containing a large number of composite fermions, yielding new quantitative and qualitative information. The ground state energy and the gaps to charged and neutral excitations are computed for a number of fractional quantum Hall effect (FQHE) states, earlier off-limits to a quantitative investigation. The ground state energies are estimated to be accurate to ~0.1% and the gaps at the level of a few percent. It is also shown that at Landau level fillings smaller than or equal to 1/9 the FQHE is unstable to a spontaneous creation of excitons of composite fermions. In addition, this approach provides new conceptual insight into the structure of the composite fermion wave functions, resolving in the affirmative the question of whether it is possible to motivate the composite fermion theory entirely within the lowest Landau level, without appealing to higher Landau levels.
Trial wave functions which, in the thermodynamic limit, describe states relevant to the fractional quantum Hall effect are explicitly constructed for up to eight electrons. Estimates are given for the ground-state energies and pair-correlation functions and for the low-lying excitation energies for states associated with the filling factors \ensuremath{\nu}=(1/3), (2/7), and (2/5). The results help explain why the filling factors \ensuremath{\nu}=${\ensuremath{\nu}}_{m}$=m/(2m+1), m=1,2,..., and \ensuremath{\nu}=1-${\ensuremath{\nu}}_{m}$ show strong quantum Hall effects.
There is increasing experimental evidence for fractional quantum Hall effect at filling factor $\ensuremath{\nu}=2+3/8$. Modeling it as a system of composite fermions, we study the problem of interacting composite fermions by a number of methods. In our variational study, we consider the Fermi sea, the Pfaffian paired state, and bubble and stripe phases of composite fermions, and find that the Fermi sea state is favored for a wide range of transverse thickness. However, when we incorporate interactions between composite fermions through composite-fermion diagonalization on systems with up to 25 composite fermions, we find that a gap opens at the Fermi level, suggesting that inter-composite fermion interaction can induce fractional quantum Hall effect at $\ensuremath{\nu}=2+3/8$. The resulting state is seen to be distinct from the Pfaffian wave function.
The composite fermion (CF) theory gives both a phenomenological description for many fractional quantum Hall (FQH) states, as well as a microscopic construction for large scale numerical calculation of these topological phases. The fundamental postulate of mapping FQH states of electrons to integer quantum Hall (IQH) states of CFs, however, was not formally established. The Landau level (LL) projection needed for the microscopic calculations is in some sense uncontrolled and unpredictable. We rigorously derive the unitary relationship between electrons and the CFs, showing the latter naturally emerge from special interactions within a single LL, without resorting to any projection by hand. In this framework, all FQH states topologically equivalent to those described by the conventional CF theory (e.g., the Jain series) have exact model Hamiltonians that can be explicitly derived, and we can easily generalize to FQH states from interacting CFs. Our derivations reveal fundamental connections between the CF theory and the pseudopotential/Jack polynomial constructions, and argue that all Abelian CF states are physically equivalent to the IQH states, while a plethora of non-Abelian CF states can be systematically constructed and classified. We also discuss about implications to experiments and effective field theory descriptions based on the descriptions with CFs as elementary particles.
Materials exhibiting exotic quantum phenomena at a macroscopic level are of great
interest because of its intrinsic value to Physics, and also because of its potential
applications in technology. Strongly correlated electronic systems form interesting
examples of these. Exploring them calls for a good understanding of its physics,
and development of new tools that are applicable to their analysis. Robustness
of the fractional quantum Hall effect (FQHE) makes it an ideal stage to study
strongly correlated electron systems. The complexity of the FQHE problem stems
directly from the strong interactions between the constituent particles and not
from the specific nature of the material it is implemented in. This work attempts
to gain a theoretical description of some of the less understood aspects of FQHE.
FQHE in the lowest Landau level is well understood using the theory of weakly
interacting emergent particles called composite fermions. Composite fermions
formed in the second Landau level however appear to be strongly interacting.
Of particular interest is the FQHE at 2 + 1/2 filling fraction, where the many body
ground state is believed to be described by a paired state of composite fermions,
represented by the Pfaffian wavefunction. Based on predicted properties of its
excitations, there has been proposals for implementing topological quantum com-
putations in this system. However a good understanding of the nature of these
excitations is still lacking.
This thesis introduces a class of many body wavefunctions called “bipartite
composite fermion states” that provide a unified description of all excitations of the
paired state at 5/2. In the bipartite composite fermion (BCF) picture, constituent
composite fermions split into two partitions which are correlated to each other.
Within each partition, they fill the lowest composite fermion Landau level of its
partition.
Properties of the BCF state can now be related to the properties of the individual partitions, which are well understood in terms of the composite fermion
theory. In particular, we show that the excitations of BCF states are obtained by
creating excitations in the individual partitions. The paired state with odd number of composite fermions should have an unpaired composite fermion. The BCF
model predicts that this unpaired composite fermion behaves as an exciton in the
quasiparticle and quasihole are in separate partitions. We confirm this prediction
numerically in finite size systems. Generalizations of bipartite composite fermion
functions to other filling fractions especially 2 + 4/7 and 2 + 3/5 are also studied.
Idea of bipartite composite fermions can be generalized to the case of more
partitions. Specifically, we look at tripartite composite fermion states which are
similar to the bipartite states, except that these have three correlated partitions
instead of two. Tripartite states have a filling fraction of ν = 3/5, and its particle
hole conjugate is a candidate wavefunction for describing the ν = 2 + 2/5…