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arXiv:cond-mat/9401002

Test of the hierarchical theory for the FQHE

Marcus Kasner

Published Jan 3, 1994• 0 citations
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Abstract

In the hierarchical theory of the fractional quantum Hall effect, the low--energy behaviour of a daughter state in the next level of the hierarchy is described by an interacting system of quasiparticles of the parent state. Taking the filled lowest Landau level as the parent state, we examine analytically the quantitative consequences of this approach for electrons interacting via a pseudopotential interaction. It is shown that the ground state energy per particle in the daughter state at a filling factor $2/3$ is exactly equal to that of a system of quasiholes in the parent state with half filling, precisely as predicted by the hierarchical approach. This is achieved with {\em only up to two--particle interactions} in the effective Hamiltonian for the quasiholes. Their single particle energy and two--particle interaction are derived. The results are generalized to the other filling factors attainable from the filled Landau level.

DOI
10.48550/arxiv.cond-mat/9401002

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In the hierarchical theory of the fractional quantum Hall effect, the low--energy behaviour of a daughter state in the next level of the hierarchy is described by an interacting system of quasiparticles of the parent state. Taking the filled lowest Landau level as the parent state, we examine analytically the quantitative consequences of this approach for electrons interacting via a pseudopotential interaction. It is shown that the ground state energy per particle in the daughter state at a filling factor 2/32/32/3 is exactly equal to that of a system of quasiholes in the parent state with half filling, precisely as predicted by the hierarchical approach. This is achieved with {\em only up to two--particle interactions} in the effective Hamiltonian for the quasiholes. Their single particle energy and two--particle interaction are derived. The results are generalized to the other filling factors attainable from the filled Landau level.
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.
I propose quasiparticle and quasihole operators which operating on the Laughlin wave functions describing the Laughlin condensates (LCs) at the filling factors ν=1/m\nu =1/mν=1/m in a specific Hilbert-subspace, {\em generate} composite fermions (CFs) by expelling electrons from the condensate to a different Hilbert-subspace. The condensation of these expelled electrons into ν=1/m\nu =1/mν=1/m together with the original LC, form a new condensate at ν=2/[2(m−1)+1]\nu = 2/[2(m- 1) + 1]ν=2/[2(m−1)+1]. In general, hierarchically constructed states are coupled LCs formed at different Hilbert-subspaces and the corresponding wave functions are {\em identical} with those proposed in the CF theory. This theory further predicts that the half and the quarter filled lowest Landau level are quantum critical points for topological phase transitions.
The Hamiltonian describing interacting two-dimensional electrons in a high magnetic field is diagonalized numerically for a small number of particles to obtain the low-lying excitation spectra. The results include estimates of energy gaps for values of $\ensuremath{\nu}$ (the lowest-Landau-level filling factor) equal to certain multiples of 15\frac{1}{5}51​, 17\frac{1}{7}71​, 19\frac{1}{9}91​, and 111\frac{1}{11}111​. These $\ensuremath{\nu}'\mathrm{s}$ are characterized by the existence of periodic rigid parent states which generate maximum phase space. The even-denominator cases are markedly different.
We present an application of collective field theory to the fractional quantum Hall effect (FQHE). We first express the condition, that the electrons are all in the lowest Landau level, as a constraint equation for the state functional. We then derive the fractional filling factor from this equation together with the no-free-vortex assumption. A hierarchy of filling factors is derived by using the particle-vortex dual transformations. In the final section we discuss an attempt at a dynamical theory of FQHE, which would justify the no-free-vortex assumption. A derivation of Laughlin’s wave function with and without quasi-hole excitations is also given.
Motivated by the structure of the quasiparticle wavefunction in the composite fermion (CF) theory for fractional quantum Hall filling factor [Formula: see text] (m odd), I consider a suitable quasiparticle operator in differential form, as a modified form of Laughlin's quasiparticle operator, that reproduces quasiparticle wave function as predicted in the CF theory, without a priori assumption of the presence of CFs. I further consider the conjugate of this operator as quasihole operator for obtaining a novel quasihole wave function for 1/m state. Each of these wave functions is interpreted as expelling an electron into a different Hilbert subspace from the original Hilbert space of the Laughlin condensate while still maintaining its correlation (although changed) with the electrons in the condensate such that the expelled electron behaves as a CF with respect to the electrons in the condensate. With this interpretation, I show that the ground state wavefunctions for general states at filling fractions [Formula: see text], respectively, can be constructed as coherent superposition of n coupled Laughlin condensates and their 'conjugates', formed at different Hilbert subspaces. The corresponding wave functions, specially surprising for [Formula: see text] sequence of states, are identical with those proposed in the theory of noninteracting CFs. The states which were considered as fractional quantum Hall effect of interacting CFs, can also be treated in the same footing as for the prominent sequences of states describing as the coupled condensates among which one is a non-Laughlin condensate in a different Hilbert subspace. Further, I predict that the half filling of the lowest Landau level is a quantum critical point for phase transition between two topologically distinct phases each corresponding to a family of states: one consists of large number of coupled Laughlin condensates of filling factor 1/3 and the other corresponds to large number of coupled conjugate Laughlin condensates of filling factor 1, which may be distinguished, respectively, by the absence and presence of upstream edge modes.
It is argued that a short-range interaction induces a quasi-Landau-level structure in the filling-factor dependence of the ground-state energy of electrons in the lowest Landau level. The discussion is based on exact numerical results obtained from small-system calculations, and the adiabatic principle of Greiter and Wilczek. It is shown that by attaching a single negative flux quantum to each electron adiabatically, the noninteracting electron ground states occupying higher Landau levels are smoothly transformed into the ground states of a hard-core boson gas in the lowest Landau level. This implies a close correspondence between the noninteracting fermions and the hard-core bosons in the lowest Landau level. Numerical results suggest the possibility of a smooth transition from hard-core bosons to fermions with a short-range interaction, by continuously varying the interaction potential.