Two-dimensional quantum dots in high magnetic fields: rotating-electron-molecule versus composite-fermion approach

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Quantum dots in high magnetic fields: Rotating-Wigner-molecule versus

composite-fermion approach

Constantine Yannouleas∗ and Uzi Landman†

School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430

(Dated: April 2003)

Exact diagonalization results are reported for the lowest rotational band of N = 6 electrons instrong magnetic fields in the range of high angular momenta 70 ≤ L ≤ 140 (covering the correspond-ing range of fractional filling factors 1/5 ≥ ν ≥ 1/9). A detailed comparison of energetic, spectral,and transport properties (specifically, magic angular momenta, radial electron densities, occupa-tion number distributions, overlaps and total energies, and exponents of current-voltage power law)shows that the recently discovered rotating-electron-molecule wave functions [Phys. Rev. B 66,115315 (2002)] provide a superior description compared to the composite-fermion/Jastrow-Laughlinones.

PACS numbers: 73.21.La; 71.45.Gm; 71.45.Lr; 73.23.-b

I. INTRODUCTION

Two-dimensional (2D) N -electron systems (with asmall finite N) in strong magnetic fields (B) have beenthe focus of extensive theoretical investigations in the lasttwenty years.1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17 The princi-pal motivations for these research activities are: (I) Theearly realization1,2 that certain special states of few elec-tron systems are relevant18 through appropriate analo-gies to the physics of the fractional quantum Hall ef-fect (FQHE), observed in the infinite 2D electron gas;(II) The unavoidable necessity, due to computer limi-tations, to test proposed model wave functions for theFQHE through numerical calculations for finite-size sys-tems; (III) The recent progress in nanofabrication tech-niques at semiconductor interfaces that has allowed ex-periments on 2D quantum dots (QD’s), with refined con-trol of their size, shape, and number of electrons19,20,21

(down to a few electrons).The physics of such systems (i.e., QD’s in high B),

is most often described with the use of composite-fermion4/Jastrow-Laughlin1 (CF/JL) analytic trial wavefunctions in the complex plane. However, it is well knownthat the thematic framework of the CF/JL approachis built on the so-called Jastrow correlations associatedwith a particular short-range interparticle repulsion.22 Ina recent paper,15 using as a thematic basis the pictureof collectively rotating electron (or Wigner) molecules(REM’s), we have derived a different class of analyticand parameter-free trial wave functions. The promisingproperty of these REM wave functions is that, unlike theCF/JL ones, they capture the all-important correlationsarising from the long-range character of the Coulombforce.

In this paper, we present an in-depth assessment ofthe CF/JL and REM trial wave functions regarding theirability to approximate the exact wave functions in thecase of QD’s (this case is often referred to as the “diskgeometry” in the FQHE literature). First systematic ex-act diagonalization (EXD) results are reported here forthe lowest rotational band of N = 6 electrons in strong

magnetic fields in the range of high angular momenta70 ≤ L ≤ 140 (covering the corresponding range offractional filling factors23 1/5 ≥ ν ≥ 1/9). A detailedcomparison (addressing five properties, i.e., prediction ofmagic angular momenta, radial electron densities, occu-pation number distributions, overlaps and total energies,and exponents of current-voltage power law) shows thatthe REM wave functions yield a superior description tothat obtained through the composite-fermion/Jastrow-Laughlin ones.

The plan of this paper is as follows: Section II presentsan outline of the REM theory, while section III focuses ona brief review of the composite-fermion approach. Exact-diagonalization results and comparisons with the CF/JLand REM wave functions are presented in section IV.Finally, our results are summarized in section V.

II. OUTLINE OF REM THEORY

In the last eight years, and in particular since1999 [when it was demonstrated24 that Wigner crys-tallization is related to symmetry breaking at theunrestricted Hartree-Fock (UHF) mean-field level],the number of publications8−17,24−40 addressing theformation and properties of Wigner (or electron)molecules in 2D QD’s and quantum dot moleculeshas grown steadily. A consensus has been reachedthat rotating electron molecules are formed bothin zero12,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40 andhigh8,9,10,11,12,13,14,15,16,17 magnetic fields.

At B = 0, formation of REM’s in QD’s is analogousto Wigner crystallization in infinite 2D media, i.e., whenthe strength of the interelectron repulsion relative to thezero-point kinetic energy (RW ) exceeds a certain criti-cal value, electrons spontaneously crystallize around sitesforming geometric molecular structures. At high mag-netic fields, the formation of Wigner molecules may bethought of as involving a two-step crystallization process:(I) the localization of electrons results from the shrinkageof the orbitals due to the increasing strength of the mag-

2

netic field; (II) then, even a weak interelectron Coulombrepulsion is able to arrange the localized electrons accord-ing to geometric molecular structures (thus this process isindependent of the value of RW ). It has been found8,10,12

that the molecular structures at high B coincide with theequilibrium configurations at B = 0 of N classical pointcharges.41,42

Due to the finite number, N , of electrons, however,there are two crucial differences between the REM andthe bulk Wigner crystal. Namely, (I) the crystallinestructure is that of the equilibrium 2D configuration ofN classical point charges, and thus consists of nestedpolygonal rings;43 (II) the Wigner molecules rotate as awhole (collective rotations) in analogy with the case of3D natural molecules.

A most striking observation concerning the REM’sis that their formation and properties have been es-tablished with the help of traditional ab initio many-body methods, i.e., exact diagonalization,9,10,11,16,25,27,38

quantum Monte Carlo26,29,33,39 (QMC), and the system-atic controlled hierarchy8,12,15,17,24,34,35,40 of approxima-tions involving the UHF and subsequent post-Hartree-Fock methods. This contrasts with the case of the CF/JLwave functions, which were inspired through “intuition-based guesswork”.

In spite of its firm foundation in many-body theory,however, the REM picture has not, until recently, suc-cessfully competed with the CF/JL picture; indeed manyresearch papers44,45,46,47,48,49,50,51 and books18 describethe physics of QD’s in high magnetic fields followingexclusively notions based on CF/JL functions, as ex-pounded in 1983 (see Ref. 1) and developed in detail in1995 in Ref. 6 and Ref. 7. We believe that one of themain obstacles for more frequent use of the REM picturehas been the lack of analytic correlated wave functionsassociated with this picture. This situation, however, haschanged with the recent explicit derivation of such REMwave functions.15

The approach used in Ref. 15 for constructing theREM functions in high B consists of two-steps: First thebreaking of the rotational symmetry at the level of thesingle-determinantal unrestricted Hartree-Fock approxi-mation yields states representing electron molecules (orfinite crystallites, also referred to as Wigner molecules,see Ref. 24 and Ref. 12). Subsequently the rotation ofthe electron molecule is described through restorationof the circular symmetry via post Hartree-Fock meth-ods, and in particular Projection Techniques.52 Natu-rally, the restoration of symmetry goes beyond the singledeterminantal mean-field description and yields multi-determinantal wave functions. For QD’s, we have shownthat the method of symmetry restoration is applicable toboth the zero34,40 and high15 magnetic-field cases.

In the zero and low-field cases, the broken symmetryUHF orbitals need to be determined numerically, and,in addition, the restoration of the total-spin symmetryneeds to be considered for unpolarized and partially po-larized cases. The formalism and mathematical details

of this procedure at B = 0 have been elaborated in Ref.34 (see also Ref. 53 and Ref. 54) for the restoration ofthe total spin in the case of quantum dot molecules).

In the case of high magnetic fields, one can specifi-cally consider the limit when the confining potential canbe neglected compared to the confinement induced bythe magnetic field. Then, assuming a symmetric gauge,the UHF orbitals can be represented15,55 by displacedGaussian analytic functions, centered at different posi-tions Zj ≡ Xj + ıYj according to the equilibrium config-uration of N classical point charges41,42 arranged at thevertices of nested regular polygons (each Gaussian rep-resenting a localized electron). Such displaced Gaussiansare written as (here and in the following ı ≡

√−1)

u(z , Zj) = (1/√π)

× exp[−|z − Zj |2/2] exp[−ı(xYj − yXj)], (1)

where the phase factor is due to the gauge invariance.z ≡ x + ıy (see Ref. 56), and all lengths are in dimen-

sionless units of lB√

2 with the magnetic length beinglB =

√

hc/eB.

In Ref. 15, we used these analytic orbitals to first con-struct the broken symmetry UHF determinant, ΨUHF

N ,and then proceeded to derive analytic expressions forthe many-body REM wave functions by applying ontoΨUHF

N an appropriate projection operator15 OL that re-stores the circular symmetry and generates correlated57

wave functions with good total angular momentum L.These REM wave functions can be easily written down15

in second-quantized form for any classical polygonalring arrangement (n1, n2, ...) by following certain simplerules for determining the coefficients of the determinantsD(l1, l2, ..., lN ) ≡ det[zl1

1 , zl22 , · · ·, zlN

N ], where the lj ’s de-note the angular momenta of the individual electrons.Since we will focus here on the case of N = 6 and N = 3electrons, we list for completeness the REM functions as-sociated with the (0, N) and (1, N−1) ring arrangements,respectively [here (0, N) denotes a regular polygon withN vertices, such as an equilateral triangle or a regularhexagon, and (1, N − 1) is a regular polygon with N − 1vertices and one occupied site in its center],

ΦL(0, N) =

l1+···+lN=L∑

0≤l1<l2<···<lN

(

N∏

i=1

li!

)−1

×

∏

1≤i<j≤N

sin[ π

N(li − lj)

]

× D(l1, l2, ..., lN ) exp(−N∑

i=1

ziz∗i /2), (2)

with

L = L0 +Nm, m = 0, 1, 2, 3, ..., (3)

3

and

ΦL(1, N −1) =

l2+···+lN=L∑

1≤l2<l3<···<lN

(

N∏

i=2

li!

)−1

×

∏

2≤i<j≤N

sin

[

π

N − 1(li − lj)

]

× D(0, l2, ..., lN) exp(−N∑

i=1

ziz∗i /2), (4)

with

L = L0 + (N − 1)m, m = 0, 1, 2, 3, ..., (5)

where L0 = N(N − 1)/2 is the minimum allowed totalangular momentum for N (polarized) electrons in highmagnetic fields.

Notice that the REM wave functions [Eq. (2) and Eq,(4)] vanish identically for values of the total angular mo-menta outside the specific values given by Eq. (3) andEq. (5), respectively.

III. OUTLINE OF COMPOSITE-FERMION

THEORY

According to the CF picture,6 the many body wavefunctions in high magnetic fields that describe N -electrons in the disc geometry (case of 2D QD’s) are givenby the expression,

ΦCFL (N) = PLLL

∏

1≤i<j≤N

(zi − zj)2mΨIPM

L∗ , (6)

where z = x+ıy and ΨIPML∗ is the Slater determinant of N

non-interacting electrons of total angular momentum L∗;it is constructed according to the Independent ParticleModel (IPM) from the Darwin-Fock58 orbitals ψp,l(z),where p and l are the number of nodes and the angularmomentum, respectively [for the values of p and l in thenth Landau level in high B, see the paragraph followingequation (7) below].

The Jastrow factor in front of ΨIPML∗ is introduced to

represent the effect of the interelectron Coulombic inter-action. In the CF literature, this assumption is oftendescribed by saying that “the Jastrow factor binds 2mvortices to each electron of ΨIPM

L∗ to convert it into acomposite fermion”.

The single-particle electronic orbitals in the Slater de-terminant ΨIPM

L∗ are not restricted to the lowest Landaulevel (LLL). As a result, it is necessary to apply a pro-jection operator PLLL to guarantee that the CF wavefunction lies in the LLL, as appropriate for B → ∞.

Since the CF wave function is an homogeneous poly-nomial in the electronic positions zj ’s, its angular mo-mentum L is related to the non-interacting total angularmomentum L∗ as follows,

L = L∗ +mN(N − 1) = L∗ + 2mL0. (7)

There is no reason to a priori restrict the Slater de-terminants ΨIPM

L∗ to a certain form, but according toRef. 6, such a restriction is absolutely necessary in or-der to derive systematic results. Thus following Ref.6, henceforth, we will restrict the non-interacting L∗ tothe range −L0 ≤ L∗ ≤ L0, and we will assume thatthe Slater determinants ΨIPM

L∗ are the so-called com-pact ones. Let Nn denote the number of electrons inthe nth Landau Level (LL) with

∑t

n=0Nn = N ; t isthe index of the highest occupied LL and all the lowerLL’s with n ≤ t are assumed to be occupied. Thecompact determinants are defined as those in which theNn electrons occupy contiguously the single-particle or-bitals (of each nth LL) with the lowest angular momenta,l = −n,−n + 1, ...,−n + Nn − 1 [p + (|l| − l)/2 = n].The compact Slater determinants are usually denoted as[N0, N1, ..., Nt], and the corresponding total angular mo-

menta are given by L∗ = (1/2)∑t

s=0Ns(Ns − 2s− 1).Most important for our present study is the fact that

the Jastrow-Laughlin wave functions with angular mo-mentum L = (2m + 1)L0 [corresponding to fractionalfilling factors ν = L0/L = 1/(2m+ 1)],

ΦJLL (N) =

∏

1≤i<j≤N

(zi − zj)2m+1 exp

(

−N∑

k=1

zkz∗k/2

)

,

(8)are a special case of the CF functions for L∗ = L0, i.e.,

ΦJLL (N) = ΦCF

L (N ;L∗ = L0), L = (2m+ 1)L0. (9)

Note that for L∗ = L0, all the non-interacting electronsoccupy contiguous states in the LLL (n = 0) with l =0, 1, ..., N − 1.

The CF/JL wave functions [equations (6) and (8)] arerepresented by compact, one-line mathematical expres-sions, which however are not the most convenient forcarrying out numerical calculations. Numerical studies ofthe CF/JL functions usually employ sophisticated MonteCarlo computational techniques. The REM wave func-tions, on the other hand, are by construction expressed insecond-quantized (superposition of Slater determinants)form, precisely like the wave functions from exact diag-onalization, a fact that greatly simplifies the numericalwork. In the numerical calculations involving JL wavefunctions in this paper, we have circumvented the needto use Monte Carlo techniques, since we were able to de-termine the Slater decomposition59 of the JL states withthe help of the symbolic language MATHEMATICA.60

We stress again that, unlike the REM functions, theCF/JL wave functions have not been derived microscop-ically, i.e., from the many-body Schrodinger equationwith interelectron Coulombic repulsions. Attempts havebeen made to justify them a posteriori by pointing outthat their overlaps with exact wave functions are close tounity or that their energies are close to the exact ener-gies. However, we will show below that this agreementis limited to rather narrow ranges of filling factors be-tween 1 ≥ ν ≥ 1/3 or to small electron numbers N ; as

4

soon as one extends the comparisons to a broader rangeof ν’s for N ≥ 6, as well as to other quantities like elec-tron densitiess and occupation number distributions, thisagreement markedly deteriorates.

IV. EXACT DIAGONALIZATION RESULTS

AND COMPARISONS

In the case of high magnetic fields, the Hilbert spacefor exact-diagonalization calculations can be restrictedto the LLL and many such calculations have beenreported2,3,5,9,10,18,44,46,49,61,62,63,64 in the past twentyyears. However, for N ≥ 5, such EXD studies havebeen restricted to angular momenta corresponding to therather narrow range of fillings factors 1 ≥ ν ≥ 1/3.

In this paper, we have performed systematic EXD cal-culations in the LLL for N = 6 electrons covering themuch broader range of fillings factors 1 ≥ ν ≥ 1/9; sucha range corresponds to angular momenta 15 ≤ L ≤ 140(note that for ν = 1/3 one has L = 45). Of crucial im-portance for extending the calculations to such large L’shas been our use of Tsiper’s65 analytic formula for cal-culating the two-body matrix elements of the Coulombinterelectron repulsion; this formula expresses the ma-trix elements as finite sums of positive terms. Earlieranalytic formulas3 suffered from large cancellation errorsdue to summations over alternating positive and negativeterms. At the same time, Tsiper’s formula is computa-tionally faster compared to the slowly-convergent seriesof Ref. 61.

For the solution of the large scale, but sparse, Coulombeigenvalue problem, we have used the ARPACK com-puter code.66 For a given L, the Hilbert space is builtout of Slater determinants,

D(l1, l2, ..., lN) exp(−N∑

i=1

ziz∗i /2), (10)

with

l1 < l2 < ... < lN ,

N∑

k=1

lk = L, (11)

and its dimensions are controlled by the maximum al-lowed single-particle angular momentum lmax, such thatlk ≤ lmax, 1 ≤ k ≤ N . We have used lmax = lJL

max + 5 =10(m+ 1) (see Ref. 59 for the definition of lJL

max) for eachgroup of angular momenta L corresponding to the range1/(2m + 1) ≤ ν < 1/(2m − 1), m = 1, 2, 3, 4. For ex-ample, for L = 105, lmax = 40 and the dimension of theHilbert space is 56115; for L = 135, lmax = 50 and thesize of the Hilbert space is 187597. By varying lmax, wehave checked that this choice produces well convergednumerical results.

A. Predictions of magic angular momenta

Foe N = 6, Figs. 1-4 display (in four installments) thetotal interaction energy from EXD as a function of thetotal angular momentum L in the range 19 ≤ L ≤ 140.(The total kinetic energy, being a constant, can be dis-regarded.) One can immediately observe the appearanceof downward cusps, implying states of enhanced stability,at certain “magic angular momenta”.

For the CF theory, the magic angular momenta can bedetermined by Eq. (7), if one knows the non-interactingL∗’s; the CF magic L’s in any interval 1/(2m − 1) ≥ν ≥ 1/(2m + 1) [15(2m − 1) ≤ L ≤ 15(2m + 1)], m =1, 2, 3, 4, ..., can be found by adding 2mL0 = 30m unitsof angular momentum to each of the L∗’s. To obtain thenon-interacting L∗’s, one needs first to construct6,9 thecompact Slater determinants. The compact determinantsand the corresponding non-interacting L∗’s are listed inTable I.

There are nine different values of L∗’s, and thus theCF theory for N = 6 predicts that there are always ninemagic numbers in any interval 15(2m−1) ≤ L ≤ 15(2m+1) between two consecutive JL angular momenta 15(2m−1) and 15(2m + 1), m = 1, 2, 3, ... (henceforth we willdenote this interval as Im). For example, using Table Iand Eq. (7), the CF magic numbers in the interval 15 ≤L ≤ 45 (m = 1) are found to be the following nine,67

15, 21, 25, 27, 30, 33, 35, 39, 45. (12)

On the other hand, in the interval 105 ≤ L ≤ 135 (m =4), the CF theory predicts the following set of nine magicnumbers,

105, 111, 115, 117, 120, 123, 125, 129, 135. (13)

An inspection of the total-energy-vs.-L plots in Figs.1-4 reveals that the CF prediction badly misses the ac-tual magic angular momenta specified by the EXD cal-culations as those associated with the downward cusps.

TABLE I: Compact non-interacting Slater determinants andassociated angular momenta L∗ for N = 6 electrons accord-ing to the CF presciption. Both L∗ = −3 and L∗ = 3 areassociated with two compact states each, the one with lowestenergy being the preferred one.

Compact state L∗

[1,1,1,1,1,1] −15[2,1,1,1,1] −9[2,2,1,1] −5[3,1,1,1] −3[2,2,2] −3[3,2,1] 0[4,1,1] 3[3,3] 3[4,2] 5[5,1] 9[6] 15

5

2.6

3

3.4

3.8

4.2

4.6

10 20 30 40 50

En

erg

y

L

FIG. 1: Total interaction energy from exact-diagonalizationcalculations as a function of the total angular momentum(10 ≤ L ≤ 50) for N = 6 electrons in high magnetic field. Theupwards pointing arrows indicate the magic angular momentacorresponding to the classically most stable (1,5) polygonalring arrangement of the Wigner molecule. The short down-wards pointing arrows indicate successful predictions of thecomposite-fermion model. The long downward arrow indi-cates a magic angular momentum not predicted by the CFmodel. Energies in units e2/κlB , where κ is the dielectricconstant.

Indeed it is immediately apparent that the number ofdownward cusps in any interval Im is always differentfrom 9. Indeed, there are 10 cusps in I1 (including thatat L = 15, not shown in Fig. 1), 10 in I2 (see Fig. 2), 7 inI3 (see Fig. 3), and 7 in I4 (see Fig. 4). In detail, the CFtheory fails in the following two aspects: (I) There areexact magic numbers that are consistently missing fromthe CF prediction in every interval; with the exception ofthe lowest L = 20, these exact magic numbers (markedby a long downward arrow in the figures) are given byL = 10(3m − 1) and L = 10(3m + 1), m = 1, 2, 3, 4, ...;(II) There are CF magic numbers that do not correspondto downward cusps in the EXD calculations (marked bymedium-size downward arrows in the figures). This hap-pens because cusps associated with L’s whose differencefrom L0 is divisible by 6 (but not simultaneously by 5)progressively weaken and completely disappear in the in-tervals Im with m ≥ 3; only cusps with the differenceL − L0 divisible by 5 survive. On the other hand, theCF model predicts the appearance of four magic num-bers with L − L0 divisible solely by 6 in every intervalIm, at L = 30m ∓ 9 and 30m ∓ 3, m = 1, 2, 3, ... Theoverall extent of the inadequacy of the CF model canbe appreciated better by the fact that there are six false

2.1

2.3

2.5

2.7

2.9

3.1

40 50 60 70 80

En

erg

y

L

FIG. 2: Total interaction energy from exact-diagonalizationcalculations as a function of the total angular momentum(40 ≤ L ≤ 80) for N = 6 electrons in high magnetic field. Theupwards pointing arrows indicate the magic angular momentacorresponding to the classically most stable (1,5) polygo-nal ring arrangement of the Wigner molecule. The shortdownwards pointing arrows indicate successful predictions ofthe composite-fermion model. The medium-size downwardspointing arrow indicates a prediction of the CF model thatfails to materialize as a magic angular momentum. The longdownward arrows indicate magic angular momenta not pre-dicted by the CF model. Energies in units of e2/κlB , whereκ is the dielectric constant.

predictions (long and medium-size downward arrows) inevery interval Im with m ≥ 3, compared to only five cor-rect ones (small downward arrows, see Fig. 3 and Fig.4).

In contrast to the CF model, the magic angular mo-menta in the REM theory are associated with the polygo-nal ring configurations of N classical point charges. Thisis due to the fact that the enhanced stability of the down-ward cusps results from the coherent collective rotation ofthe regular-polygon REM structures. Due to symmetryrequirements, such collective rotation can take place onlyat magic-angular-momenta values. The in-between angu-lar momenta require the excitation of additional degreesof freedom (like the center of mass and/or vibrationalmodes), which raises the total energy with respect to thevalues associated with the magic angular momenta.

For N = 6, the lowest in energy ring configurationis the (1,5), while there exists a (0,6) isomer41,42 withhigher energy. As a result, our EXD calculations (as wellas earlier ones9,11,13 for lower angular momenta L ≤ 70)have found that there exist two sequences of magic an-gular momenta, a primary one (Sp) with L = 15 + 5m

6

1.8

2

2.2

70 80 90 100 110

Energ

y

L

FIG. 3: Total interaction energy from exact-diagonalizationcalculations as a function of the total angular momentum(70 ≤ L ≤ 110) for N = 6 electrons in high magneticfield. The upwards pointing arrows indicate the magic an-gular momenta corresponding to the classically most stable(1,5) polygonal ring arrangement of the Wigner molecule.The short downwards pointing arrows indicate successful pre-dictions of the composite-fermion model. The medium-sizedownwards pointing arrows indicate predictions of the CFmodel that fail to materialize as magic angular momenta. Thelong downward arrows indicate magic angular momenta notpredicted by the CF model. Energies in units of e2/κlB , whereκ is the dielectric constant.

[see Eq. (3)], associated with the most stable (1,5) classi-cal molecular configuration, and a secondary one (Ss)with L = 15 + 6m [see Eq. (5)], associated with themetastable (0, 6) ring arrangement. Furthermore, ourcalculations (see also Refs. 11,13) show that the sec-ondary sequence Ss contributes only in a narrow range ofthe lowest angular momenta; in the region of higher an-gular momenta, the primary sequence Sp is the only onethat survives and the magic numbers exhibit a period offive units of angular momentum. It is interesting to notethat the initial competition between the primary and sec-ondary sequences, and the subsequent prevalence of theprimary one, has been seen in other sizes as well,11 i.e.,N = 5, 7, 8. Furthermore, this competition is reflectedin the field-induced molecular phase transitions associ-ated with broken symmetry UHF solutions in a parabolicQD. Indeed, Ref. 17 demonstrated recently that, as afunction of increasing B, the UHF solutions for N = 6first depict the transformation of the maximum-density-droplet68 into the (0,6) molecular configuration; then (athigher B) the (1,5) configuration replaces the (0,6) struc-ture as the one having the lower HF energy.69

1.6

1.7

1.8

1.9

100 110 120 130 140

Energ

y

L

FIG. 4: Total interaction energy from exact-diagonalizationcalculations as a function of the total angular momentum(100 ≤ L ≤ 140) for N = 6 electrons in high magneticfield. The upwards pointing arrows indicate the magic an-gular momenta corresponding to the classically most stable(1,5) polygonal ring arrangement of the Wigner molecule.The short downwards pointing arrows indicate successful pre-dictions of the composite-fermion model. The medium-sizedownwards pointing arrows indicate predictions of the CFmodel that fail to materialize as magic angular momenta. Thelong downward arrows indicate magic angular momenta notpredicted by the CF model. Energies in units of e2/κlB , whereκ is the dielectric constant.

The extensive comparisons in this subsection lead in-evitably to the conclusion that the CF model cannot ex-plain the systematic trends exhibited by the magic angu-lar momenta in 2D QD’s in high magnetic fields. Thesetrends, however, were shown to be a natural consequenceof the formation of REM’s and their metastable isomers.

B. Radial electron densities

We turn now our attention to a comparison of the ra-dial electron densities (ED’s). Fig. 5 displays the corre-sponding ED’s from EXD, REM, and CF/JL wave func-tions at three representative total angular momenta, i.e.,L = 75 (ν = 1/5), 105 (1/7), and 135 (1/9).

An inspection of Fig. 5 immediately reveals that (I)The EXD radial ED’s (solid lines) exhibit a prominentoscillation corresponding to the (1, 5) molecular struc-ture (averaged over the azimuthal angles). Indeed theintegral of the exact ED’s from the origin to the mini-mum point between the two humps is practically equalto unity; (II) There is very good agreement between

7

0

0.4

0.8

0

0.4

0.8

0

0.4

0.8

0 2 4 6 8 10

r (l ) B

2pr(

r)

(l

)

B

-2

L=75

L=105

L=135

JL

FIG. 5: Radial electron densities for N = 6 electrons inhigh magnetic field. Solid line: densities from exact diago-nalization. Dashed line: densities from REM wave functions.Dotted line: densities from Jastrow-Laughlin wave functions.

the REM (dashed lines) and exact ED’s; this agreementimproves with higher angular momentum; (III) The JLED’s (dotted lines ) miss the oscillation of the exact EDin all three cases in a substantial way.

The inability of the radial ED’s calculated with the JLfunctions to capture the oscillations exhibited by the ex-act ones was also seen recently for the ν = 1/3 case andfor all electron numbers N = 6, 7, 8, 9, 10, 11, 12 in Ref. 62(see in particular Fig. 1 therein). We further note thatthe oscillations of the exact ED’s in that figure corre-spond fully to the classical molecular ring arrangementslisted in Ref. 41, e.g., to (1,7) for N = 8 and to (3,9)for N = 12, in agreement with our rotating-electron-molecule interpretation.

0

0.4

0.8

0

0.4

0.8

0

0.4

0.8

0 10 20 30 40

L=75

L=105

L=135

Occu

pa

tio

n n

um

be

r, n

Single-particle angular momentum, l

JL

FIG. 6: Distribution of occupation numbers as a functionof single-particle angular momentum l for N = 6 electronsin high magnetic field. Solid circles: occupation numbersfrom exact diagonalization. Open circles: occupation num-bers from REM wave functions. Crosses: occupation numbersfrom Jastrow-Laughlin wave functions.

C. Distribution of occupation numbers

In this subsection, we address the behavior of the

occupation-number distribution n(l) = 〈Φ|a†lal|Φ〉 asa function of the single-particle angular momentum l,where the creation and annihilation operators refer tothe single-electron states ψ0,l(z) in the LLL. For N = 6,Fig. 6 displays the n(l)’s from all three families of wavefunctions, i.e., EXD (solid circles), REM (open circles),and JL (crosses), and for the three representative angularmomenta L = 75 (ν = 1/5), 105 (1/7), and 135 (1/9).

Again, an inspection of Fig. 6 immediately reveals that(I) The EXD occupation numbers exhibit a prominent os-cillation corresponding to the (1, 5) molecular structure.Indeed the sum of the exact n(l)’s from l = 0 to the min-imum point between the two humps is practically equal

8

TABLE II: Case of N = 3 electrons in high mag-netic fileds. Overlaps, 〈ΦL|Φ

EX

L 〉/(〈ΦL|ΦL〉〈ΦEX

L |ΦEX

L 〉)1/2, ofREM’s (Φ’s) and JL functions (Φ’s) with the correspondingexact eigenstates (ΦEX’s) for various values of the angular mo-menta L (ν are the corresponding fractional filling factors).Recall that the angular momenta for the JL functions areLJL = N(N − 1)(2m + 1)/2, with m = 0, 1, 2, 3, ... The JLoverlaps are from Ref. 1.

L(ν) JL REM9(1/3) 0.99946 0.98347

15(1/5) 0.99468 0.9947321(1/7) 0.99476 0.9967427(1/9) 0.99573 0.99758

33(1/11) 0.99652 0.9980739(1/13) 0.99708 0.99839

to unity; (II) There is very good agreement between theREM and exact occupation numbers; this agreement im-proves with higher angular momentum; (III) For all threecases, the JL occupation numbers exhibit a systemati-cally different trend and they are not able to capture theoscillatory behavior of the EXD occupation numbers.

We further note that a substantial discrepancy betweenJL and EXD occupation numbers was also noted in Ref.64 for the case of N = 7 electrons and ν = 1/3 (L = 63).

The systematic deviations between the JL and EXDED’s and occupation numbers inevitably points to theconclusion that these two families of wave functions rep-resent very different many-body physical problems. In-deed, the JL functions have been found22 to be exact so-lutions for a special class of short-range two-body forces,while the EXD functions faithfully reflect the long-rangecharacter of the Coulombic interelectron repulsion. Onthe other hand, as discussed in Ref. 15, the REM wavefunctions, derived through a traditional many-body ap-proach, are able to capture the correlations arising fromthe long-range character of the Coulomb force; the os-cillatory behavior of the EXD and REM ED’s and oc-cupation numbers (associated with formation of Wignermolecules) constitutes a prominent and unmistaken sig-nature of such Coulombic correlations.

TABLE III: Overlaps of JL and REM wave functions with theexact ones for N = 6 electrons and various angular momentaL (ν are the corresponding fractional filling factors).

L(ν) JL REM75(1/5) 0.837 0.817

105(1/7) 0.710 0.850135(1/9) 0.665 0.860

D. Comparison of overlaps and total energies

We turn now our attention to the overlaps of the REMand JL wave functions with those obtained through ex-act diagonalization. We start by listing in Table II theoverlaps for the simpler case of N = 3 electrons in highmagnetic fields. One sees immediately that these over-laps are all very close to unity (≥ 0.99) for both the REMand JL cases and for even rather high angular momenta[e.g., L = 39 (ν = 1/13)].

Ever since they were calculated by Laughlin in his orig-inal paper,1 the JL overlaps for N = 3 electrons have ex-ercised a great influence in the literature of the fractionalquantum Hall effect (FQHE). Indeed, in a rather sweep-ing generalization to any N and L (note that Ref. 62 hasindeed found that the JL overlaps for ν = 1/3 remainvery close to unity for all cases with 5 ≤ N ≤ 12), theclose-to-unity values of the JL overlaps have been pre-sumed to provide “proof” that the CF/JL functions ap-proximate very well the corresponding exact many-bodywave functions; as we have already shown earlier, thispresumption is highly questionable.

We have calculated the overlaps for N = 6 elec-trons and for the three representative higher angular-momentum values L = 75 (ν = 1/5), 105 (1/7), and135 (1/9); the results are listed in Table III for both theREM and JL wave functions. A most remarkable featureof the results in Table III is that the extraordinary, higherthan 0.99 values (familiar from Laughlin’ s paper1) aretotally absent. Instead, the JL overlaps rapidly deteri-orate for higher L’s (lower ν’s), and for ν = 1/9 theyhave attained values below 0.67. In contrast, the REMoverlaps remain above 0.80 and slowly approach unity asL increases.

From our results for ν ≤ 1/5 and the results of Ref.62 for ν = 1/3, it is apparent that the overlaps aloneare not a reliable index for assessing the agreement ordisagreement between trial and exact wave functions. Forexample, for N = 6 and L = 75 (ν = 1/5), Table IIIshows that the JL and REM overlaps are close to eachother (0.837 vs. 0.817). However, as the earlier analysesbased on the electron densities and occupation numbersshow, the JL wave function is not a good approximationto the exact one; in contrast, the REM wave functionoffers a much better description.

In addition to the overlaps, earlier studies (see, e.g.,Ref. 44) have also relied on the total energies for assess-ing the agreement, or not, between CF and exact wavefunctions. We thus list in Table IV the total energies forN = 6 and for the three representative higher angular-momentum values L = 75 (ν = 1/5), 105 (1/7), and135 (1/9). It is seen that both the JL and REM totalenergies exhibit very small relative errors compared tothe corresponding EXD ones in all three instances, a factthat indicates that, by themselves, the total energies70

are an even less reliable index compared to the overlaps.In particular, note that for N = 6 and L = 135, the JLand exact total energies differ only in the third decimal

9

TABLE IV: Total interaction energies of JL, REM, and exact-diagonalization wave functions for N = 6 electrons and var-ious angular momenta L (ν are the corresponding fractionalfilling factors). The percentages within parentheses indicaterelative errors. Recall that the angular momenta for the JLfunctions are LJL = N(N − 1)(2m + 1)/2, m = 0, 1, 2, 3, ...Energies in units of e2/κlB , where κ is the dielectric constant.

L(ν) JL REM EXACT75(1/5) 2.2093 (0.32%) 2.2207 (0.85%) 2.201885(3/17) 2.0785 (0.65%) 2.065195(3/19) 1.9614 (0.55%) 1.9506105(1/7) 1.8618 (0.46%) 1.8622 (0.48%) 1.8533115(3/23) 1.7767 (0.45%) 1.7692125(3/25) 1.7020 (0.38%) 1.6956135(1/9) 1.6387 (0.50%) 1.6361 (0.34%) 1.6305

point, while at the same time the JL overlap is only 0.665(see Table III)!

E. Exponents of current-voltage power law

Another quantity of theoretical and experimental in-terest is the ratio

α =n(lJL

max − 1)

n(lJLmax)

, (14)

of the corresponding occupation numbers at lJLmax − 1

and lJLmax. The interest in this ratio is due to the fol-

lowing two facts: (I) The value of α for the JL func-tion at different fractional fillings has a particular an-alytic value,71,72,73 i.e., it is given by αJL(ν) = 1/ν =2m+ 1, m = 1, 2, 3, 4, ...; (II) α happens to enter as theexponent72,73 of the voltage in the current-voltage law,I ∝ V α, for external electron tunneling into an edge ofa fractional quantum Hall system. Recent investigationshave found that both the experimental74 and computed73

EXD value of α at ν = 1/3 deviates from the JL predic-tion of 3, being in all instances somewhat smaller (i.e.,∼ 2.7).

Table V displays the values of α for N = 6 and forthe JL, REM, and EXD wave functions at various val-ues of the total angular momentum L. We have checkedthat our numerical values for αJL (derived by dividingthe proper nJL’s; see Fig. 6) are equal to 2m+ 1 withinthe numerical accuracy. As seen from Table V, a moststriking weakness of the JL functions is that the cor-responding αJL’s diverge as L → ∞, a behavior whichcontrasts sharply with the EXD values that remain atall times finite and somewhat smaller than 3. Such adramatic difference in behavior should be possible to bechecked experimentally. Furthermore, we note that theREM values, although somewhat smaller, they are closeto the EXD ones and remain bounded as L→ ∞.

We conclude that this dramatic qualitative and quan-titative weakness of the JL functions is due to their be-ing exact solutions of a family of short range interparticle

TABLE V: Values of the ratio α [Eq. (14)] for JL, REM,and exact-diagonalization wave functions for N = 6 electronsand various angular momenta L; ν (given in parentheis) arethe corresponding fractional filling factors. Recall that theangular momenta for the JL functions are LJL = N(N −1)(2m + 1)/2, m = 0, 1, 2, 3, ...

L(ν) JL REM EXACT75(1/5) 5.000 1.964 2.877

105(1/7) 7.000 1.972 2.708135(1/9) 9.000 1.978 2.726

forces.22 On the other hand, as we have stressed earlier inthis paper and in Ref. 15, the REM functions are able tocapture the essential effects of the correlations associatedwith the long-range Coulomb force; thus, in agreementwith the EXD results, the REM α values remain finite asL→ ∞.

V. SUMMARY

Exact diagonalization (EXD) results for the lowestrotational band of a circular QD with N = 6 elec-trons in strong magnetic fields were reported75 here forthe first time in the range of high angular momenta70 ≤ L ≤ 140 (covering the corresponding range of frac-tional filling factors 1/5 ≥ ν ≥ 1/9). These EXD re-sults were used in a thorough assessment of the ability ofthe composite-fermion4/Jastrow-Laughlin1 and rotating-electron-molecule15 trial wave functions to approximatethe exact wave functions in the case of 2D QD’s.

A detailed comparison (addressing five properties, i.e.,prediction of magic angular momenta, radial electrondensities, occupation number distributions, overlaps andtotal energies, and exponents of current-voltage powerlaw) shows that the REM many-body wave functionsprovide a description that is superior to that obtainedthrough the CF/JL ones. An important finding is that“global” quantities (like overlaps and total energies) arenot particularly reliable indices for comparing exact andtrial wave functions; a reliable decision on the agree-ment, or lack of it, between exact and trial wave func-tions should include detailed comparisons of quantitieslike radial electron densities and/or occupation numberdistributions.

We finally note that the CF/JL wave functions havebeen most useful for the modeling of the bulk fractionalquantum Hall effect. However, the theoretical investiga-tions concerning the bulk system have unavoidably, dueto computational limitations, relied on finite-size systemsto assess the validity of the CF/JL wave functions. Thusit is natural to conjecture that the unexpected finding ofthis paper, i.e., that the CF/JL functions exhibit remark-able weaknesses in reproducing the exact wave functionsof QD’s in high B, may have ramifications for our presentunderstanding of the fractional quantum Hall effect it-

10

self. Investigations of such probable ramifications, andrelated questions concerning the domain of validity of theREM and CF/JL wave functions in the bulk, will be ad-dressed in future publications. In the present paper, wefocused on the case of QD’s, which constitute a theoret-ically self-contained problem when exact-diagonalizationcalculations become available; in the near future, a wider

range of such calculations will be within reach, due tonew generations of powerful computers.

This research is supported by the U.S. D.O.E. (GrantNo. FG05-86ER-45234). Computations were carried outat the Georgia Tech Center for Computational Materi-als Science and the National Energy Research ScientificComputing Center (NERSC).

∗ Electronic address: [email protected]† Electronic address: [email protected] R.B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).2 R.B. Laughlin, Phys. Rev. B 27, 3383 (1983).3 S.M. Girvin and T. Jach, Phys. Rev. B 28, 4506 (1983).4 J.K. Jain, Phys. Rev. B 41, 7653 (1990).5 S.-R. Eric Yang, A.H. MacDonald, and M.D. Johnson,

Phys. Rev. Lett. 71, 3194 (1993).6 J.K. Jain and T. Kawamura, Europhys. Lett. 29, 321

(1995).7 T. Kawamura and J.K. Jain, J. Phys.: Condens. Matter 8,

2095 (1996).8 H.-M. Muller and S.E. Koonin, Phys. Rev. B 54, 14532

(1996).9 T. Seki, Y. Kuramoto, and T, Nishino, J. Phys. Soc. Jpn.

65, 3945 (1996).10 P.A. Maksym, Phys. Rev. 53, 10871 (1996).11 W.Y. Ruan and H-F. Cheung, J. Phys.: Condens. Matter

11, 435 (1999).12 C. Yannouleas and U. Landman, Phys. Rev. B 61, 15895

(2000).13 P.A. Maksym, H. Imamura, G.P. Mallon, and H. Aoki, J.

Phys.: Condens. Matter 12, R299 (2000).14 C.E. Creffield, J.H. Jefferson, S. Sarkar, and D.L.J. Tipton,

Phys. Rev. B 62, 7249 (2000).15 C. Yannouleas and U. Landman, Phys. Rev. B 66, 115315

(2002).16 M. Rontani, G. Goldoni, F. Manghi, and E. Molinari, Eu-

rophys. Lett. 58, 555 (2002).17 B. Szafran, S. Bednarek, and J. Adamowski, Phys. Rev. B

67, 045311 (2003).18 See, e.g., A. L. Jacak, P. Hawrylak, and A. Wojs, Quantum

Dots (Springer, Berlin, 1998), in particular Ch. 4.5.19 R.C. Ashoori, Nature (London) 379, 413 (1996).20 S. Tarucha, D.G. Austing, T. Honda, R.J. van der Hage,

and L.P. Kouwenhoven, Phys. Rev. Lett. 77, 3613 (1996).21 L.P. Kouwenhoven, C.M. Marcus, P.L. McEuen, S.

Tarucha, R.M. Westervelt, and N.S. Wingreen, Proceed-ings of the NATO Advanced Study Institute on Meso-

scopic Electron Transport , Series E, Vol. 345, edited byL.L. Sohn, L.P. Kouwenhoven, and G. Schon (Kluwer, Dor-drecht, 1997) p. 105.

22 F.D.M. Haldane, Phys. Rev. Lett. 51, 605 (1983); S.A.Trugman and S. Kivelson, Phys. Rev. B 31, 5280 (1985).

23 We use the well-known formula ν = N(N −1)/2L (see Ref.2), which specifies the corresponding fractional filling fac-tors in the thermodynamic limit. We stress, however, thatin this paper we focus exclusively on finite-size systems;thus, throughout this paper, ν is used as a more compactindex in place of L.

24 C. Yannouleas and U. Landman, Phys. Rev. Lett. 82, 5325(1999); ibid. 85, 2220(E) (2000).

25 C.E. Creffield, W. Hausler, J.H. Jefferson, and S.Sarkar,Phys. Rev. B 59, 10719 (1999).

26 R. Egger, W. Hausler, C.H. Mak, and H. Grabert, Phys.Rev. Lett 82, 3320 (1999); ibid. 83, 462(E) (1999).

27 C. Yannouleas and U. Landman, Phys. Rev. Lett. 85, 1726(2000).

28 W. Hausler, B. Reusch, R. Egger, and H. Grabert, PhysicaB 284, 1772 (2000).

29 A.V. Filinov, Y.E. Lozovik, and M. Bonitz, Phys. StatusSolidi B 221, 231 (2000).

30 S.M. Reimann, M. Koskinen, and M. Manninen, Phys.Rev. B 62, 8108 (2000).

31 B. Reusch, W. Hausler, and H. Grabert, Phys. Rev. B 63,113313 (2001).

32 A. Matulis and F.M. Peeters, Solid State Commun. 117,655 (2001).

33 A.V. Filinov, M. Bonitz, and Y.E. Lozovik, Phys. Rev.Lett. 86, 3851 (2001).

34 C. Yannouleas and U. Landman, J. Phys.: Condens. Mat-ter 14, L591 (2002).

35 P.A. Sundqvist, S.Y. Volkov, Y.E. Lozovik, and M. Wil-lander, Phys. Rev. B 66, 075335 (2002).

36 S.A. Mikhailov and K. Ziegler, Eur. Phys. J. B 28, 117(2002).

37 S.A. Mikhailov, Physica E 12, 884 (2002).38 S.A. Mikhailov, Phys. Rev. B 65, 115312 (2002).39 A. Harju, S. Siljamaki, and R.M. Nieminen, Phys. Rev. B

65, 075309 (2002).40 C. Yannouleas and U. Landman, arXiv:

cond-mat/0302130.41 V.M. Bedanov and F.M. Peeters, Phys. Rev. B 49, 2667

(1994).42 F. Bolton and U. Rossler, Superlatt. Microstruct. 13, 139

(1993).43 Under conditions of partial spin polarization (i.e., low mag-

netic fields), the molecular configurations may exhibit dis-tortions away from the classical equilibrium configurations.With increasing RW , however, the classical molecular con-figurations are recovered (see Ref. 38).

44 J.K. Jain and R.K. Kamilla, Int. J. Mod. Phys. B 11, 2621(1997).

45 E. Goldmann and S.R. Renn, Phys. Rev. B 56, 13296(1997).

46 S.-R. Eric Yang and J.H. Han, Phys. Rev. B 57, R12681(1998).

47 M. Taut, J. Phys.: Condens. Matter 12, 3689 (2000).48 M. Manninen, S. Viefers, M. Koskinen, and S.M. Reimann,

Phys. Rev. B 64, 245322 (2001).49 X. Wan, K. Yang, and E.H. Rezayi, Phys. Rev. Lett. 88,

056802 (2002).50 X. Wan, E.H. Rezayi, and K. Yang, ArXiv:

cond-mat/0302341.

11

51 A. Harju, S. Siljamaki, and R.M. Nieminen, Phys. Rev.Lett. 88, 226804 (2002).

52 P. Ring and P. Schuck, The Nuclear Many-body Problem

(Springer, New York, 1980) Ch. 11, and references therein.53 C. Yannouleas and U. Landman, Eur. Phys. J. D 16, 373

(2001).54 C. Yannouleas and U. Landman, Int. J. Quantum Chem.

90, 699 (2002).55 K. Maki and X. Zotos, Phys. Rev. B 28, 4349 (1983).56 The definition z ≡ x+ıy is associated with positive angular

momenta for the single-particle states in the lowest Landaulevel. In Ref. 15, we used z ≡ x − ıy and negative single-particle angular momenta in the lowest Landau level. Thefinal expressions for the trial wave functions do not dependon these choices.

57 Both the finite-size REM molecule and several sophisti-cated bulk Wigner crystal (BWC) approaches at high B(listed at the end of this footnote) start with a single-determinantal UHF wave function constructed out of theorbitals in Eq. (1), and both do improve it by introducingadditional correlations; however, the nature of these corre-lations is quite different between the REM and the BWCapproaches. Indeed, due the the finite-size of the system,the REM approach includes correlations associated withfluctuations in the azimuthal angle (see Ref. 15 and Ref.40); these correlations arise from the restoration of thecircular symmetry and result in states with good total an-gular momenta (in particular magic angular momenta, seesection IV.A). Naturally, in the BWC approaches, angular-momentum conservation and magic angular momenta arenot considered; for example, Lam and Girvin include cor-relations from vibrational -type fluctuations of the BWCthat are more in tune with the expected translational in-variance of a bulk system. As a result, the REM exhibitsdrastically different properties from the properties of an N-electron piece of the bulk Wigner crystal. Rather, the REMwave functions exhibit properties associated with the in-compressible magic-angular-momenta states in the spectraof QD’s, which are finite-size precursors to the “correlated-liquid” fractional quantum Hall states of the bulk [see Ref.18]. For sophisticated BWC approaches at high B, see, e.g.,P.K. Lam and S.M. Girvin, Phys. Rev. B 30, 473 (1984);H. Yi and H.A. Fertig, Phys. Rev. B 58, 4019 (1998).

58 C.G. Darwin, Proc. Cambridge Philos. Soc. 27, 86 (1930);V. Fock, Z. Phys. 47, 446 (1928).

59 In the case of N = 6 electrons, we have used 338, 5444,32134, and 118765 terms in this decomposition for L =45 (ν = 1/3), 75 (1/5), 105 (1/7), and 135 (1/9), re-spectively. These numbers correspond to all the Slaterdeterminants with L = 15(2m + 1) and individual an-gular momenta l ≤ lJLmax = 5(2m + 1), including thecases with zero coefficients. We remind the reader thatlJLmax = (2m+1)(N −1) is the maximum individual angularmomentum allowed in the JL states. The Slater decompo-sition of the JL states for N = 2, 3, 4, 5, 6, but only forν = 1/3, has been reported earlier in G.V. Dunne, Int. J.Mod. Phys. B 7, 4783 (1993).

60 S. Wolfram, Mathematica: A system for doing mathemat-

ics by computer (Addison-Wesley, Reading, 1991).61 M. Stone, H.W. Wyld, and R.L. Schult, Phys. Rev. B 45,

14156 (1992).62 E.V. Tsiper and V.J. Goldman, Phys. Rev. B 64, 165311

(2001).63 M. Kasner, Ann. Phys. (Berlin) 11, 175 (2002), and refer-

ences therein.64 V.A. Kashurnikov, N.V. Prokof’ev, B.V. Svistunov, and

I.S. Tupitsyn, Phys. Rev. B 54, 8644 (1996).65 E.V. Tsiper, J. Math. Phys. 43, 1664 (2002).66 R.B. Lehoucq, D.C. Sorensen, and C. Yang, ARPACK

Users’ Guide: Solution of Large-Scale Eigenvalue Prob-

lems with Implicitly Restarted Arnoldi Methods (SIAM,Philadelphia, 1998).

67 Ref. 9 gives the full list of the nine CF magic numbers inthe interval (1 ≥ ν ≥ 1/3). Ref. 6 excludes two of them,i.e., the CF magic angular momenta 27 and 33.

68 A.H. MacDonald, S.R.E. Yang, and M.D. Johnson, Aust.J. Phys. 46, 345 (1993).

69 At B = 0, the interelectron Coulombic repulsion can alsoinduce (as a function of increasing RW ) a similar successionof phase transitions [i,e., normal fluid → (0,6) molecule →(1,5) molecule], see Fig. 2 in Ref. 24.

70 There is no “variational dilemma” from the fact that theCF/JL and REM functions are two essentially differentwave functions with very close expectation values of theenergy. Indeed, the CF/JL wave functions correspond to ahamiltonian with short-range two-dody interactions, whilethe REM functions correspond to the actual hamiltonianof the Coulomb problem that involves long-range interelec-tron interactions. Therefore these represent two separatevariational problems.

71 S. Mitra and A.H. MacDonald, Phys. Rev. B 48, 2005(1993).

72 X.G. Wen, Phys. Rev. B 41, 12838 (1990); Int. J. Mod.Phys. B 6, 1711 (1992).

73 V.J. Goldman and E.V. Tsiper, Phys. Rev. Lett. 86, 5841(2001).

74 A.M. Chang, M.K. Wu, C.C. Chi, L.N. Pfeiffer, and K.W.West, Phys. Rev. Lett. 86, 143 (2001), and referencestherein.

75 We have specifically considered the limit when the con-fining potential can be neglected compared to the confine-ment induced by the high magnetic field. In high B, the ef-fect of the confining potential amounts simply in selecting aspecific magic-angular-momentum state (see section IV.A)as the ground state of the system (the specific value of themagic L depends on the strength of B and the parame-ters of the confinement). In most studies (see, e.g., Ref.6, Ref. 9, or Ref. 13), the external confinement has beenmodeled as a harmonic potential. Most recently, however,Wan et al. (Ref. 49 and Ref. 50) have studied few-electronQD’s taking into consideration a disk-like neutralizing pos-itive background. Indeed, these authors employ a confiningpotential arising from a positive background charge dis-tributed uniformly on a parallel disk at a distance d fromthe electron layer (the typical d in experiments is d ≥ 10lB ,see Ref. 50). As was the case with the harmonic externalpotential, these authors found again that their externalconfinement influences which magic-L state becomes theground state of the system. Most importantly, the ground-state wave functions in their exact-diagonalization studyexhibit strong oscillations in the radial electron density [inan apparent agreement with the classical ring configura-tions of Wigner molecules] and in disagreement with theCF/JL wave functions. It is interesting to note the coinci-dence, for all practical purposes, of the exact radial elec-tron density for N = 6 and L = 105 calulated by Wanet al. with that calculated by us [compare figure 5(d) inRef. 50 with the middle panel of figure 5 in this paper]. In

12

order to account for the disagreement between the exactand CF/JL wave functions, Wan et al. were led to use theconcept of “edge reconstruction”. In the case studied byus, however, our exact-diagonalization results (and thoseof Tsiper and Goldman, see Ref. 62) do not include anyexternal confinement, a fact that rules out “edge recon-struction” as the underlying cause for the disagreementbetween the exact and CF/JL wave functions. As we have

pointed out in this paper previously (see section IV.E andalso Ref. 15), this disagreement arises from the fact thatthe CF/JL functions do not capture the long-range charac-ter of the Coulomb interelectron repulsion. On the contrarythe REM wave functions are able to capture the long-rangeCoulombic correlations and thus are in better agreementwith the wave functions from exact diagonalization.