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International Journal of Modern Physics B

The Microscopic Model of Composite Particles for ν=1/m FQHE Edge Excitations, its Bosonization and the Chiral Condition

Wenjun Zheng

Published Sep 10, 1997• 0 citations
Publisher

Abstract

We derive a microscopic theory of the composite particles describing the low-lying edge excitations in the fractional quantum Hall liquid with ν=1/m. Using the composite-particle transformation, one finds that the edge states of the system in a disc sample are described by the Calogero–Sutherland model with other interactions between the composite particles as perturbations. The effects of both the interactions and the confinement potential are considered, and proved unable to change the exponent g=ν=1/m. We also obtain the important chiral condition at the mean field level, and as a result justify microscopically the Chiral Luttinger Liquid model of the FQHE edge states.

DOI
10.1142/s0217979297001313

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•May 1, 2012•0 citations
We derive a microscopic theory of the composite particles describing the low-lying edge excitations in the fractional quantum Hall liquid with ν=1/m. Using the composite-particle transformation, one finds that the edge states of the system in a disc sample are described by the Calogero–Sutherland model with other interactions between the composite particles as perturbations. The effects of both the interactions and the confinement potential are considered, and proved unable to change the exponent g=ν=1/m. We also obtain the important chiral condition at the mean field level, and as a result justify microscopically the Chiral Luttinger Liquid model of the FQHE edge states.
We derive a microscopic theory of the composite fermion-type quasiparticles describing the low-lying edge excitations in the fractional quantum Hall liquid with ν = 1/ m . Using the composite fermion transformation, one finds that the edge states of the system in a disc sample are described by the Calogero–Sutherland-like model (CSLM) in the one-dimensional limit. This result presents the consistency between one-dimensional and two-dimensional statistics. It is shown that the low-lying excitations, indeed, have the chiral Luttinger liquid behaviors because there is a gap between the right-and left-moving excitations of the CSLM.
Based on the composite fermion approach, we derive a microscopic theory describing the low-lying edge excitations in the fractional quantum Hall liquid with ν=ν∗ϕ~ν∗+1ν=\frac{ν^*}{\tildeϕν^*+1}ν=ϕ~​ν∗+1ν∗​. For ν∗>0ν^*>0ν∗>0, it is found that the composite fermion model reduces to an SU(ν∗)(ν^*)(ν∗) Calogero-Sutherland model in the one-dimensional limit, whereas it is not exact soluble for ν∗<0ν^*<0ν∗<0. However, the ground states in both cases can be found and the low-lying excitations can be shown the chiral Luttinger liquid behaviors since a gap exists between the right- and left-moving sectors in each branch of the azimuthal excitations.
We derive a microscopic model describing low-lying edge excitations in a fractional quantum Hall liquid with $\ensuremath{\nu}={\ensuremath{\nu}}^{*}/(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\varphi}}{\ensuremath{\nu}}^{*}+1).$ For ${\ensuremath{\nu}}^{*}>0,$ it is found that the composite-fermion model reduces to an $\mathrm{SU}({\ensuremath{\nu}}^{*})$ Calogero-Sutherland model in a dimension reduction, whereas it is not exactly soluble for ${\ensuremath{\nu}}^{*}<0.$ However, the ground states in both cases can be found, and the low-lying excitations can show chiral Luttinger-liquid behaviors. On the other hand, we show that the finite-temperature behavior of the $G\ensuremath{-}T$ curve will deviate from the prediction of the chiral Luttinger liquid. We also point out that the suppression of the ``spin'' degrees of freedom agrees with very recent experiments by Chang et al. [Phys. Rev. Lett. 77, 2538 (1996)]. The two-boson model of Lee and Wen (preprint, cond-mat/9801318) is described microscopically.
The effective field theory of the fractional quantum Hall edge is reformulated from microscopic dynamics. Noncommutative Chern-Simons theory is a microscopic description for the quantum Hall fluid. We use it for reference. Considering relabeling symmetry of the electrons and incompressibility of the fluid, we obtain a constraint and derive a chiral Tomonaga-Luttinger liquid theory containing interaction terms. We calculate one-loop corrections to the phonon and electron propagators and get a new tunneling exponent. It agrees with experiments.
We use bosonic field theories and the infinite system density matrix renormalization group method to study infinite strips of fractional quantum Hall states starting from microscopic Hamiltonians. Finite-entanglement scaling allows us to accurately measure chiral central charge, edge-mode exponents, and momenta without finite-size errors. We analyze states in the first and second levels of the standard hierarchy and compare our results to predictions of the chiral Luttinger liquid theory. The results confirm the universality of scaling exponents in chiral edges and demonstrate that renormalization is subject to universal relations in the nonchiral case. We prove a generalized Luttinger theorem involving all singularities in the momentum-resolved density, which naturally arises when mapping Landau levels on a cylinder to a fermion chain and deepens our understanding of non-Fermi liquids in one dimension.
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…