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.
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.
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…
We establish the quantum mechanics of composite fermions based on the dipole picture initially proposed by Read. It comprises three complimentary components: a wave equation for determining the wave functions of a composite fermion in ideal fractional quantum Hall states and when subjected to external perturbations, a wave-function for mapping a many-body wave function of composite fermions to a physical wave function of electrons, and a microscopic approach for determining the effective Hamiltonian of the composite fermion. The wave equation resembles the ordinary Schrödinger equation but has drift velocity corrections that are not present in the Halperin-Lee-Read theory. The wave-function constructs a physical wave function of electrons by projecting a state of composite fermions onto a half-filled bosonic Laughlin state of vortices. Remarkably, Jain's wave-function can be reinterpreted as the new in an alternative wave-function representation of composite fermions. The dipole picture and the effective Hamiltonian can be derived from the microscopic model of interacting electrons confined in a Landau level, with all parameters determined. In this framework, we can construct the physical wave function of a fractional quantum Hall state deductively by solving the wave equation and applying the wave-function , based on the effective Hamiltonian derived from first principles, rather than relying on intuition or educated guesses. For ideal fractional quantum Hall states in the lowest Landau level, the approach reproduces the well-established results of the standard theory of composite fermions. We further demonstrate that the reformulated theory of composite fermions can be easily generalized for flat Chern bands. Published by the American Physical Society 2024
Motivated by the issue of particle-hole symmetry for the composite fermion Fermi sea at the half filled Landau level, Dam T. Son has made an intriguing proposal [Phys. Rev. X {\bf 5}, 031027 (2015)] that composite fermions are Dirac particles. We ask what features of the Dirac-composite fermion theory and its various consequences may be reconciled with the well established microscopic theory of the fractional quantum Hall effect and the 1/2 state, which is based on {\em non-relativistic} composite fermions. Starting from the microscopic theory, we derive the assertion of Son that the particle-hole transformation of electrons at filling factor ν=1/2 corresponds to an effective time reversal transformation (i.e. {kj}→{−kj}) for composite fermions, and discuss how this connects to the absence of 2kF backscattering in the presence of a particle-hole symmetric disorder. By considering bare holes in various composite-fermion Λ levels (analogs of electronic Landau levels) we determine the Λ level spacing and find it to be very nearly independent of the Λ level index, consistent with a parabolic dispersion for the underlying composite fermions. Finally, we address the compatibility of the Chern-Simons theory with the lowest Landau level constraint, and find that the wave functions of the mean-field Chern-Simons theory, as well as a class of topologically similar wave functions, are surprisingly accurate when projected into the lowest Landau level. These considerations lead us to introduce a normal form for the unprojected wave functions of the n/(2pn−1) states that correctly capture the topological properties even without lowest Landau level projection.
Exact diagonalization of a two-dimensional electron gas in a strong magnetic field in the disk geometry shows that there exists a filling-factor range in the second Landau level where the states significantly differ from those in the lowest Landau level. We show that the difference arises because the interaction between composite fermions is not negligible in higher Landau levels, as indicated by a substantial mixing between composite-fermion Landau-like levels, or $\ensuremath{\Lambda}$ levels. We find that the exact ground state is well reproduced by composite-fermion theory with $\ensuremath{\Lambda}$-level mixing incorporated at the lowest level of approximation. Using the same variational approach in the spherical geometry we estimate the excitation gap at filling 1∕3 in the second Landau level (which corresponds to 7∕3 of experiment).