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Physical Review Letters

Anyonic description of low-energy physics in the fractional quantum Hall effect

Jian Cheng Yang, W. P. Su

Published Feb 22, 1993• 17 citations
Publisher

Abstract

The diagonalization of a Hamiltonian in the subspace spanned by many quasiparticle states has been performed. The result, compared to the full Hilbert space diagonalization, shows that the anyonic picture of the hierarchy theory provides a very accurate description of the low-energy physics of the fractional quantum Hall system. The implication of this study on the equivalence between the hierarchy theory and the composite fermion theory is discussed.

DOI
10.1103/physrevlett.70.1163

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The diagonalization of a Hamiltonian in the subspace spanned by many quasiparticle states has been performed. The result, compared to the full Hilbert space diagonalization, shows that the anyonic picture of the hierarchy theory provides a very accurate description of the low-energy physics of the fractional quantum Hall system. The implication of this study on the equivalence between the hierarchy theory and the composite fermion theory is discussed.
The effective partition function of a fractional quantum Hall system, which is described by the Laughlin quasiparticle wave function, is obtained in the path-integral representation. It shows that the quasiparticles cannot be treated as ordinary anyons in view of the freeze of the kinetic energy and the ``bosonic aspect'' of the Hilbert space. By using the constrained anyon systems, we find that one type of filling fractions is more stable than other fractions predicted by the standard hierarchy theory. This work also sheds considerable light on the possible equivalence between the hierarchy theory and the composite-fermion approach proposed by Jain recently.
A recently developed hierarchy theory of the fractional quantum Hall effect (FQHE) is reviewed in this article. We use a many quasiparticle (QP) wave function to obtain the effective partition function of a FQHE system in the path-integral representation, which has been shown to be formally equivalent to an anyon system confined to the lowest Landau level. However the two systems are different in the dimension of the Hilbert space, which implies that the standard hierarchy theory is invalid beyond second hierarchical states. The only way to overcome the difficulty is to place anyons in higher Landau levels. By making a mean-field approximation to the statistical gauge field, the QPs are converted to bosons and a constraint boson system is derived. We find a new type of condensate of the bosonic QPs, which results in a new type of incompressible states of the electrons. These states which involve only two hierarchies should have higher stability over others. That explains why some sequences of filling fractions occur prominently in experiment. Finally, we show that there is a one-to-one correspondence between the new hierarchy theory and the composite fermion theory proposed by Jain.
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 commutation relations of composite fields are studied in the 3, 2 and 1 spatial dimensions. It is shown that the field of an atom consisting of a nucleus field and an electron field satisfies, in the space-like asymptotic limit, the canonical commutation relations within the sub-Fock space of the atom. The field-particle duality in the bound state is discussed from a statistical point of view. Then, the commutation relations of the scalar object in the Schwinger (Thirring) model are briefly discussed and they are shown to be consistent with its interpretation as the Nambu-Goldstone boson. The composite anyon fields are shown to satisfy the proper anyonic commutation relations with additive phase exponents. Then, quasiparticle picture of the anyons is clarified under the restriction of this additivity. The difference between field and particle characteristics becomes more prominent in the 2-dimensional space. It is argued that the hierarchy of the fractional quantum Hall effect can be rather simply understood by utilizing the quasiparticle characters of the anyons in the case that the background-boson gauge is assumed. To contrast with this case, the composite fermion theories are critically reviewed.
Calculations on finite-size systems are carried out to examine the hierarchy theory of the fractional quantum Hall effect. The concept of quasiparticle and quasihole is justified to remain valid at high densities. Numerical results reveal a hard-core boundary constraint on the anyonic quasiparticles so that their wave function must vanish as two of them get close in space. The dynamics of this contraint originates from a strong short-range repulsion between the quasiparticles. The observed quasiparticle-quasihole asymmetry in the hierarchy is understood through the hard-core anyon theory.
Abstract The fractional quantum Hall effect remains a captivating area in condensed matter physics, characterized by strongly correlated topological order, which manifests as fractionalized excitations and anyonic statistics. Numerical simulations, such as exact diagonalization, density matrix renormalization groups, matrix product states, and Monte Carlo methods are essential for examining the properties of strongly correlated systems. Recently, density functional theory has been employed in this field within the framework of composite fermion theory. This paper systematically evaluates how density functional theory approaches have addressed fundamental challenges in fractional quantum Hall systems, including ground state and low-energy excitations. Special attention is given to the insights provided by density functional theory regarding composite fermion behavior, edge effects, and the nature of fractional charge and magnetoroton excitations. The discussion critically examines both the advantages and limitations of these approaches, while highlighting the productive interplay between numerical simulations and theoretical models. Future directions are explored, particularly the promising potential of time-dependent density functional theory for modeling non-equilibrium dynamics in quantum Hall systems.