Second generation of composite fermions in the Hamiltonian theory

http://doc.rero.ch

Second generation of composite fermions in the Hamiltonian theory

M. O. Goerbig,^1,2P. Lederer,²and C. Morais Smith¹

1De´partment de Physique, Universite´ de Fribourg, Pe´rolles, CH-1700 Fribourg, Switzerland

2Laboratoire de Physique des Solides, Baˆtiment 510, UPS (associe´ au CNRS), F-91405 Orsay cedex, France 共Received 4 August 2003; revised manuscript received 10 March 2004; published 22 April 2004兲 In the framework of a recently developed model of interacting composite fermions restricted to a single level, we calculate the activation gaps of a second generation of spin-polarized composite fermions. These composite particles each consist of a composite fermion of the first generation and a vortexlike excitation and may be responsible for the recently observed fractional quantum Hall states at unusual filling factors such as

␯⫽4/11, 5/13, 5/17, and 6/17. Because the gaps of composite fermions of the second generation are found to be more than one order of magnitude smaller than those of the first generation, these states are less visible than the usual states observed at filling factors␯⫽p/(2 ps⫹1). Their stability is discussed in the context of a pseudopotential expansion of the composite-fermion interaction potential.

DOI: 10.1103/PhysRevB.69.155324 PACS number共s兲: 73.43.Cd, 71.10.Pm I. INTRODUCTION

The fractional quantum Hall effect 共FQHE兲in the lowest Landau level 共LL兲 at filling factors ␯⫽p/(2 ps⫹1), with integral p and s, may be understood as an integral quantum Hall effect共IQHE兲in terms of composite fermions共CFs兲.^1,2 Each CF consists of a bound state of one electron and a vortexlike excitation with vorticity 2s and charge c²

⫽2 ps/(2 ps⫹1), in units of electron charge. The CFs expe- rience a reduced coupling (eB)*_⫽eB/(2 ps⫹1) to the ex- ternal magnetic field B and thus populate CF-LLs with a CF filling factor ␯*_⫽n_el/n_B*. Here, n_el is the electronic den- sity, and n_B*⫽1/2␲^lB*² is the density of states of each CF-LL in terms of the CF magnetic length l_B*_⫽

冑

ប/(eB)*. The CF and electronic filling factors are related by ␯

⫽␯*/(2s␯*_⫹1). The FQHE arises when␯*_⫽p: Because of the CF-LL separation, which is on the order of the char- acteristic Coulomb energy e²/⑀^lB, a finite energy is required to promote a particle to an upper CF-LL if the filling is changed. These particles become localized by residual impu- rities and therefore do not contribute to the electrical trans- port, giving rise to the observed plateaus in the Hall resis- tance.

Experiments by Pan et al. have revealed a new class of FQHE states at filling factors such as ␯⫽4/11,6/17,..., of which at least the 4/11 state is found to be spin-polarized.³ The existence of such states had been conjectured by Mani and von Klitzing based on the fractal structure and self- similarity of the Hall resistance curve.⁴It appears natural to interpret these states, found at CF filling factors ␯*_⫽1

⫹˜ /(2sp ˜ p˜⫹1), in terms of an IQHE of a second generation of CFs (C²Fs); whereas the CFs in the lowest CF-LL ( p

⫽0) remain inert, those in the partially filled p⫽1 level bind to some vortexlike excitation carrying 2s˜ additional flux quanta. This picture is reminiscent of the hierarchy scheme proposed by Haldane and Halperin, according to which Laughlin quasiparticles⁵ might in principle form an incom- pressible liquid state if the interaction potential between these quasiparticles is sufficiently short-range.^6,7 Even though the new states may be classified within this scheme,

theoretical studies seem to indicate that fully spin-polarized C²Fs and even higher generations of CFs would not be stable.^{8 –11} Exact-diagonalization studies for the 4/11 state show a gapped ground state only for 12 electrons, which is at present the maximum electron number for such numerical investigations.¹²On the other hand, a numerical diagonaliza- tion with up to 24 particles can be performed in the CF basis with a truncated Hilbert space.^10,11Within the latter formal- ism, a ground state with features of a FQHE state emerges for N⫽12 and 20 particles, but not for N⫽8, 16, and 24.¹⁰ The extrapolation to the thermodynamic limit is therefore ambiguous. Lopez and Fradkin investigated the topological stability of a FQHE state at␯⫽4/11 in a Chern-Simons field- theory approach.¹³In contrast to prior stability investigations by Haldane,¹⁴a spin-polarized state is found to be topologi- cally stable, if interpreted in terms of C²Fs.

Recently, we have derived a model of interacting CFs¹⁵in the framework of the Hamiltonian theory proposed by Mur- thy and Shankar.¹⁶ The involved transformations provide a mathematical basis for the self-similarity of the FQHE.¹⁵ Here, we construct a C²F representation of this model, in analogy with the CF representation in the Hamiltonian theory, by the restriction of the dynamics to the first excited CF-LL p⫽1. We suppose the stability of fully spin-polarized C²F states and calculate their activation gaps as a function of the finite width of the two-dimensional electron system. In contrast to prior numerical analyses in the wave-function ap- proach, the calculations are performed directly in the thermo- dynamic limit. Inter-CF-LL excitations are taken into ac- count and give rise to a screened effective interaction. The dielectric function, which is calculated in the random-phase approximation 共RPA兲, is similar to the electronic case dis- cussed by Aleiner and Glazman.¹⁷

The structure of the paper is the following: in Sec. II we present the model of interacting CFs and evaluate their ef- fective interaction potential. The dielectric function due to inter-CF-LL excitations is derived in Sec. III. In Sec. IV we construct the C²F basis and calculate the activation gaps of the C²F states as a function of the finite width of the electron system. The resulting activation gap for␯⫽4/11 is compared to the experimental estimate. The stability of C²F states is

Published in "Physical Review B 69: 115324, 2004"

which should be cited to refer to this work.

http://doc.rero.ch

discussed in terms of a pseudopotential expansion of the CF interaction potential in Sec. V. A brief summary may be found in Sec. VI.

II. MODEL

We adopt a model of spin-polarized particles, the dynam- ics of which are restricted to the lowest electronic LL. The low-energy degrees of freedom are given by the Hamiltonian

Hˆ⫽1

2

兺

_q ^{v0共q兲␳¯共⫺q兲␳¯共q兲, 共1兲}

where v0(q)⫽(2␲^e2/⑀^q)exp(⫺q²l_B²/2), with l_B⫽

冑

ប/eB.

Instead of representing the projected density operators^{¯ (q)}␳ in the usual electron basis, an alternative representation in terms of CFs is chosen in the Hamiltonian theory of the FQHE: a ‘‘preferred’’ combination of the projected density and the density¯ (q) of the vortexlike object is introduced,␹

␳

¯_CF(q)⫽␳^{¯ (q)}⫺c²␹^{¯ (q).16}This preferred combination plays the role of the CF density operator and may be expressed as

␳

¯_CF共q兲⫽_n,n⬘

兺

_;m,m⬘^{具m兩e⫺iq•RCF兩m⬘典}

⫻具^n兩¯␳^p共q兲兩n⬘典^cn,m

† c_n⬘,m⬘,

where c_n,m^† creates a CF in the n-th CF-LL with the guiding- center quantum number m, associated with the CF guiding center coordinate R_CF. The matrix elements are 共for m

⭓m⬘⁾

具^m兩e^⫺iq•RCF兩m⬘典⫽

冑

^mm!⬘!

冉

^{⫺i共qx&⫹iqy兲lB*}

冊

^m⫺m⬘

⫻L_m^m_⬘^⫺m⬘

冉

^q22lB*2

冊

^{e⫺q2lB*2/4, 共2兲}

and共for n⭓n⬘⁾

具ⁿ兩^¯␳^p共q兲兩n⬘典⫽

冑

^nn!⬘!

冉

^{⫺i共qx⫺&iqy兲lB*c}

冊

^{n⫺n⬘e⫺q2lB*2c2/4}

⫻

冋

^{Lnn⬘⫺n⬘}

冉

^q2lB*22c2

冊

^{⫺c2共1⫺n⫹n⬘兲}

⫻e^⫺q2lB2/2c2L_nⁿ_⬘^⫺n⬘

冉

^q2c2lB*22

冊册

^{, 共3兲}

with the associated Laguerre polynomials L_n^m(x) and the CF magnetic length l_B*_⫽l_B/

冑

¹⫺c².

In order to describe the low energy degrees of freedom in the experimentally relevant range of CF fillings 1⬍␯*_⬍2, we restrict␳^¯CF(q) to the CF-LL p⫽1 in the same manner as for electrons. The restricted CF-density operator is 具^¯␳CF(q)典p⫽1⫽具¹兩^¯␳^p(q)兩1典␳ញ^{(q), with}

␳ញ共q兲⫽_m,m

兺

_⬘^{具m兩e⫺iq•RCF兩m⬘典cn†⫽1,mcn⫽1,m}⬘.

The model Hamiltonian of restricted CFs therefore becomes¹⁵

H˜共s兲⫽1

2

兺

_q ^{˜v共q兲␳ញ共⫺q兲␳ញ共q兲, 共4兲}

where the CF form factor

F_CF共q兲⬅具^1兩¯␳^p共q兲兩1典

⫽e^{⫺q2lB*2c2/4}

冋

^L1

冉

^q2lB*22c2

冊

⫺c²e^⫺q2lB2/2c2L₁

冉

^q2c2lB*22

冊册

has been absorbed into an effective CF interaction potential

v

˜共q兲⫽v0共q兲关F_CF共q兲兴²

⑀共q兲 . 共5兲 Note that the derivation of this CF model is not limited to p⫽1, but may also be applied for higher CF-LLs. In this case the CF form factor depends on p, F_CF^p (q)

⫽具^p兩¯␳^p(q)兩p典. The projected CF-density operators satisfy the same commutation relations

关␳ញ共q兲,␳ញ共k兲兴⫽2i sin

冉

^{共q⫻k2兲zlB*2}

冊

^{␳ញ共q⫹k兲}

as the projected density operators in the model of electrons restricted to the lowest LL if one replaces the electronic by the CF magnetic length.^16,18This and the fact that the Hamil- tonian of interacting CFs 共4兲 has the same structure as the original electronic one¹⁵ allows us treat the model with the same theoretical tools which have been used for the under- standing of the FQHE at␯⫽p/(2 ps⫹1). One would simply need to replace the electronic by the CF interaction potential, v₀(q)→˜ (q), and take into account the renormalization ofv the magnetic length, l_B→l_B*.

III. DIELECTRIC FUNCTION

In contrast to the electronic case, inter-CF-LL excitations have to be considered explicitly in a dielectric function⑀^(q), which modifies the CF interaction potential, because the re- sidual CF interactions are intrinsically on the same order of magnitude as the CF-LL separation. Both energy scales are given by the Coulomb interaction e²/⑀^lB.

For the calculation of the dielectric function, we investi- gate inter-CF-LL excitations in the original Hamiltonian共1兲 in the CF basis. Its solution in the Hartree-Fock approxima- tion yields the ‘‘free’’ CF Hamiltonian

Hˆ

CF

0 ⫽

兺

n,m ␧nc_n,m^† c_n,m. 共6兲 It can be shown that the CF-LLs are approximately linear in n, ⑀n⯝n␻C*, where ␻C* is the energy of a quasiparticle ex- citation in units of ប⬅1, and we have omitted an unimpor-

http://doc.rero.ch

tant constant. The corrections beyond the Hartree-Fock ap- proximation are given by the ‘‘interaction’’ Hamiltonian

Hˆ

CF int⫽1

2_␯

兺

1,...,␯4

v_␯

1,...,␯₄关c_␯

1

† c_␯

2

† c_␯

4c_␯

3⫺具^c_␯^†₁^c␯₃典^c_␯^†₂^c␯₄

⫺具^c␯†₂c_␯

4典^c␯†₁c_␯

3⫹具^c␯†₁c_␯

4典^c␯†₂c_␯

3⫹具^c␯†₂c_␯

3典^c␯†₁c_␯

4兴, 共7兲 with␯i⫽(n_i,m_i) and the interaction vertex

v_␯

1,...,␯₄⫽

兺

_q ^{v0共q兲具n1兩␳¯p共⫺q兲兩n3典具n2兩¯␳p共q兲兩n4典}

⫻具^m1兩e^iq•RCF兩m₃典具^m2兩e^⫺iq•RCF兩m₄典^. The special form of the interaction Hamiltonian 共7兲leads to the omission of all diagrams containing the equal-time con- tractions 具^Tcn,m(␶^)cn,m

† (␶⁾典 in a diagrammatic perturbation expansion, whereTdenotes time-ordering. The averages are taken over the Hartree-Fock ground state characterized by 具^cn,m

† c_n⬘,m⬘典⫽␦n,n⬘␦m,m⬘⌰( p⫺1⫺n), where ⌰(x) is the step function with ⌰(x)⫽0 for x⬍0 and ⌰(x)⫽1 for x

⭓0. The dielectric function may be calculated in the RPA,

⑀R PA(q,␻⁾⫽1⫺v₀(q)D(q,␻^{), where}

D共q,␻兲⫽⫺i

冕

^{dtei␻t具T¯␳CF共q,t兲¯␳CF共⫺q,t⫽0兲典}

⫽2n

兺

⬎0 Fn

p共q兲 n␻C*

␻²⫺n²␻C*² is the density-density Green’s function with

Fn

p共q兲⫽_n⬘^p_⫽

兺

^⫺_p¹_⫺n ^{兩具n⬘⫹n兩␳¯p共q兲兩n⬘典兩2. 共8兲}

For lower p, the static dielectric function in the limit␻→0 is a finite sum over Laguerre polynomials, which may be di- rectly calculated with the help of the matrix elements共3兲. In the case of p⫽1 one finds

⑀R PA共p⫽1兲共q兲⫽1⫹v0共q兲共ql_B*c兲²

␻C*

冋

^L1

冉

^q2lB*22c2

冊

⫺e^⫺q2lB2/2c2L₁

冉

^q2c2lB*22

冊册

^{2e⫺q2lB*2c2/2.}

For larger values of p, one may use the asymptotic form of the amplitudes共8兲

Fn

p共q兲⯝n

冋

^Jn共qlB*c

^冑

^{2 p⫹1兲⫺c2Jn}

冉

^qlcB*

^冑

^{2 p⫹1}

冊册

^2,

in terms of the Bessel functions J_n(x). This yields the static dielectric function

⑀R PA

p 共q兲⯝1⫹v0共q兲

␻C*

再

^{1⫹c4⫺2c2J0}

^冋

^qRC*

^冉

^c⫺1c

^冊册

⫺

冋

^{J0共qRC*c兲⫺c2J0}

冉

^qRcC*

冊册

²

冎

^{, 共9兲}

with the CF cyclotron radius R_C*_⫽l_B*

冑

^{2 p}⫹1. This is similar to the sreening of the effective electron-electron interaction in higher electronic LLs.¹⁷Note that the RPA is more prob- lematic in the CF than in the electronic case because (e²/⑀^lB)/␻C* is not a physically small parameter. Therefore, other diagrams may also play a role in the dielectric function.

IV. C²F BASIS AND ACTIVATION GAPS

In order to obtain the C²F representation of the model, we proceed in the same manner as for the construction of the CF representation.¹⁶A new preferred combination is introduced,

␳

¯_C2F(q)⫽␳ញ^(q)⫺˜c²␹ញ^{(q), where} ␹ញ(q) describes a vortex- like excitation of the CF liquid with vorticity 2s˜ and charge c

˜²⫽2 p˜ s˜/(2 p˜ s˜⫹1) in units of the CF charge e*_⫽1⫺c². This density operator becomes in the C²F basis, with obvious notation

␳

¯_C2F共q兲⫽_n,n⬘

兺

_;m,m⬘ ^{具m兩e⫺iq•RC2F兩m⬘典}

⫻具ⁿ兩␳ញ^p共q兲兩n⬘典^˜cn,m

† ˜c_n⬘,m⬘,

where c˜_n,m^† creates a C²F in the C²F-LL n in the guiding- center state m. The degeneracy of each C²F-LL is now n_B^˜

⫽1/2␲^{l˜2, where l˜}⫽l_B*/

冑

¹⫺˜c²⫽l_B/

冑

⁽¹⫺c²)(1⫺˜c²) is the C²F magnetic length. The matrix elements are given by Eqs. 共2兲and共3兲if one replaces l_B*_→l˜ and c→˜ .c

The activation gap for C²Fs is calculated in the same manner as for CFs;¹⁶ it is the sum of the energies of a qua- siparticle and a quasihole excitation ⌬^a(s;s˜, p˜ )

⫽⌬^{q p}(s;s˜, p˜ )⫹⌬^qh(s;s˜, p˜ ), with

⌬^{q p}共s;s˜, p˜兲⫽具^˜cp˜ ,mH˜共s兲˜c_{˜ ,mp}^† 典⫺具^H˜典 and

⌬^qh共s;s˜, p˜兲⫽具^˜c˜p⫺1,m

† H˜共s兲˜c_p˜⫺1,m典⫺具^H˜典^.

With the help of Wick contractions, one averages over the C²F ground state, which consists of p˜ completely filled C²F-LLs and may thus be characterized by 具^˜cn,m

† ˜c_n⬘,m⬘典

⫽␦n,n⬘␦m,m⬘⌰( p˜⫺1⫺n). This yields the activation gaps

⌬^a共s;s˜, p˜兲⫽1

2

兺

_q ^{v˜共q兲关具˜p兩␳ញp共⫺q兲␳ញp共q兲兩˜p典}

⫺具^˜p⫺1兩␳ញ^p共⫺q兲␳ញ^p共q兲兩˜p⫺1典兴

⫺

兺

_q ^˜v共q兲˜_n^p

兺

_⫽^⫺₀^{1 关兩具˜p兩␳ញp共q兲兩n典兩2}

⫺兩具^˜p⫺1兩␳ញ^p共q兲兩n典兩²兴, in analogy with the case of CFs.¹⁶

http://doc.rero.ch

In units of e²/⑀^lB and for characteristic quasiparticle en- ergies␻C*found in the litterature,^1,16,19the activation gaps of some chosen states with s⫽1 and p⫽1 共with␻C*_⫽0.1) read as

whereas for states with s⫽2 and p⫽1 共with␻C*_⫽0.03), one obtains

Note that the C²F states at␯*_⫽1⫹˜ /(2 pp ˜ s˜⫹1) and␯*_⫽2

⫺˜ /(2 pp ˜ s˜⫹1) are related by the particle-hole symmetry in the same manner as the CF states at ␯⫽p/(2 ps⫹1) and ␯

⫽1⫺p/(2 ps⫹1). The 4/11 state is therefore equivalent to the states at ␯⫽5/13, 7/11, and 8/13.

Finite width effects may be included by replacing the original interaction potential v₀(q)→v₀(q) f (␭,q), where f (␭,q)⫽exp(␭²q²)关1⫺Erf(␭q)兴 is given in terms of the er- ror function Erf(x). This expression has been obtained under the assumption that the confining potential in the z direction 共with a characteristic width ␭兲 be quadratic.^16,19The results are shown in Fig. 1 for different filling factors. As expected from the FQHE of electrons, the activation gap is reduced for samples with larger ␭.¹⁹

The only activation gap of this series which was inferred from experiments is the one at␯⫽4/11, which is on the order of 50 mK for a sample width of⬃500 Å.³This corresponds to a width parameter ␭⯝6.3l_B at B⫽10 T. The activation gap obtained theoretically for this width is ⌬^a(s⫽1;s˜⫽1,p˜

⫽1)⯝0.0025e²/⑀^lB, which is on the order of 400 mK and thus one order of magnitude larger than the experimental activation gap and almost three times larger than the value calculated by Chang et al. for a partially spin-polarized state.¹¹Two effects may account for this discrepancy. First, it is known that the Hamiltonian theory for the conventional FQHE overestimates the activation gap by a factor of⬃1.5 in comparison with numerical studies even if the agreement becomes better for wider samples.¹⁶This effect is expected to be also present in the C²F formalism. A second and more likely origin of the discrepancy between the theoretical and experimental results is the fact that the theory does not ac- count for impurity effects. They lead to a gap reduction, which may be on the same order of magnitude as the activa- tion gap in the absence of impurities.²⁰ Impurities affect these sensitive C²F states more than their corresponding CF states because the activation gap of the latter is more than one order of magnitude larger.

V. CRITERION FOR THE STABILITY OF C²FS

Finally, we comment on the stability of C²Fs. In order to conclude about their existence in nature, one

would in principle have to compare the energy of the C²F liquid with possibly competing phases, e.g., solid phases of CFs in p⫽1.²¹ However, insight into their stability may be obtained from an expansion of the CF inter- action potential 共5兲 in Haldane’s pseudopotentials,⁶ V_m^CF⫽兺q˜ (q)Lv _m(q²l_B²*)exp(⫺q²l_B²*/2), with odd m. The re- sults for p⫽1 and different s, i.e., for the CF interaction in the filling-factor range 1/(2s⫹1)⬍␯⬍2/(4s⫹1), are shown in Fig. 2. In contrast to the electronic case, one ob- serves a non-monotonic behavior for the unscreened CF in- teraction 共black lines兲; the largest value is found for m⫽1, but there is a large suppression of the component V₃^CF, FIG. 1. Activation gaps as a function of width parameter for different fillings.共a兲s⫽1: ␯⫽4/11共black circles兲,␯⫽6/17共white circles兲,␯⫽7/19共black squares兲, and␯⫽11/31共white squares兲;共b兲 s⫽2: ␯⫽4/19 共black circles兲, ␯⫽6/29 共white circles兲, ␯⫽7/33 共black squares兲, and␯⫽11/53共white squares兲.

FIG. 2. Haldane’s pseudopotentials for CF interaction both with- out screening共black line兲 and with screening共gray line兲; the lines are a guide to the eyes.

http://doc.rero.ch

which is smaller than V₅^CF and even becomes negative for larger s. A similar result has been obtained in previous stud- ies in the wave-function approach.^8,21 The non-monotonic behavior is likely to be due to the dipolelike internal struc- ture of the CF.^16,22,23An attractive dipole-dipole interaction is thus superimposed on the repulsive Coulomb repulsion.

This stabilizes the C²F states at␯*_⫽1⫹1/3, such as the 4/11 state, which have their largest weight in the m⫽1 angular- momentum component, but may destabilize states at ␯*_⫽1

⫹1/5, e.g., the one at␯⫽6/17. If one takes into account the screening, given by the dielectric function 共9兲, the compo- nent V₃^CF increases, and the potential becomes more Coulomb-like共gray lines兲. The 4/11 state is found to remain stable in the presence of screening because for s⫽1 the ratio V₁^CF/V₃^CF⫽1.56 remains sufficiently high, i.e., the potential is sufficiently short-range. A first-order phase transition to a different state is expected to appear for a ratio V₁/V₃ⱗ1.2.²⁴

VI. CONCLUSIONS

In conclusion, we have interpreted the experimentally ob- served FQHE³at␯⫽4/11 as an IQHE of a second generation of CFs (C²Fs) in the framework of a recently developed model of interacting CFs,¹⁵ which is based on the Hamil- tonian theory of the FQHE.¹⁶After restriction of the dynam- ics to the CF-LL p⫽1, the effective Hamiltonian for C²Fs

becomes similar to the CF Hamiltonian in the lowest elec- tronic LL. The formalism is generic and may be applied for other CF-LLs, p⬎1, as well as for higher generations of CFs.¹⁵ Screening effects due to inter-CF-LL excitations, which in contrast to their electronic counterpart cannot be neglected even in the large-B limit, are included in a dielec- tric function, calculated in the RPA. Both screening and the finite width of the two-dimensional electron gas give rise to a modification of the effective interaction potential and lead to a reduction of the C²F activation gaps, which are about one order of magnitude smaller than those for CFs. The the- oretical gap at␯⫽4/11 is, however, substantially larger than the gap inferred from the experiment.³ This discrepancy is likely to be due to residual impurities in the sample, which have been omitted in the theoretical model and which are known to produce a further reduction of the activation gaps.²⁰ Whereas the stability of a fully polarized 4/11 state remains controversial,8 –10,13,14

a pseudopotential expansion of the CF interaction potential hints to a stable 4/11 state if interpreted in terms of C²Fs. The definitive answer to this question, however, requires detailed energy calculations in- volving all possible competing states, namely CF Wigner crystal, CF bubble, and C²F liquid phases.

We acknowledge fruitful discussions with R. Morf. This work was supported by the Swiss National Foundation for Scientific Research under Grant No. 620-62868.00.

1J. K. Jain, Phys. Rev. Lett. 63, 199共1989兲; Phys. Rev. B 41, 7653 共1990兲.

2Perspectives in Quantum Hall Effects, edited by S. Das Sarma and A. Pinczuk共Wiley, New York, 1997兲.

3W. Pan, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, Phys. Rev. Lett. 90, 016801共2003兲.

4R. G. Mani and K. v. Klitzing, Z. Phys. B: Condens. Matter 100, 635共1996兲.

5R. B. Laughlin, Phys. Rev. Lett. 50, 1395共1983兲.

6F. D. Haldane, Phys. Rev. Lett. 51, 605共1983兲.

7B. I. Halperin, Phys. Rev. Lett. 52, 1583共1984兲.

8P. Be´ran and R. Morf, Phys. Rev. B 43, 12 654共1991兲.

9A. Wo´js and J. J. Quinn, Phys. Rev. B 61, 2846共2000兲.

10S. S. Mandal and J. K. Jain, Phys. Rev. B 66, 155302共2002兲.

11C.-C. Chang, S. S. Mandal, and J. K. Jain, Phys. Rev. B 67, 121305共2003兲.

12R. Morf共private communication兲.

13A. Lopez and E. Fradkin, cond-mat/0310128共unpublished兲.

14F. D. Haldane, Phys. Rev. Lett. 74, 2090共1995兲.

15M. O. Goerbig, P. Lederer, and C. Morais Smith, cond-mat/

0401340共unpublished兲.

16G. Murthy and R. Shankar, Rev. Mod. Phys. 75, 1101共2003兲; R.

Shankar, Phys. Rev. B 63, 085322共2001兲.

17I. L. Aleiner and L. I. Glazman, Phys. Rev. B 52, 11 296共1995兲.

18S. M. Girvin, A. H. MacDonald, and P. M. Platzman, Phys. Rev.

B 33, 2481共1986兲.

19R. Morf, N. d’Ambrumenil, and S. Das Sarma, Phys. Rev. B 66, 075408共2002兲.

20R. Morf and N. d’Ambrumenil, Phys. Rev. B 68, 113309共2003兲.

21S.-Y. Lee, V. W. Scarola, and J. K. Jain, Phys. Rev. Lett. 87, 256803共2001兲; Phys. Rev. B 66, 085336共2002兲.

22N. Read, Semicond. Sci. Technol. 9, 1859共1994兲.

23V. Pasquier and F. D. Haldane, Nucl. Phys. B 516, 719共1998兲.

24D. Yoshioka, The Quantum Hall Effect共Springer, Berlin, 2002兲.