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Saarikoski, Henri; Reimann, S.M.; Räsänen, Esa; Harju, Ari; Puska, MarttiStability of vortex structures in quantum dots

Published in:Physical Review B

DOI:10.1103/PhysRevB.71.035421

Published: 26/01/2005

Document VersionPublisher's PDF, also known as Version of record

Please cite the original version:Saarikoski, H., Reimann, S. M., Räsänen, E., Harju, A., & Puska, M. (2005). Stability of vortex structures inquantum dots. Physical Review B, 71(3), 1-7. [035421]. https://doi.org/10.1103/PhysRevB.71.035421

Stability of vortex structures in quantum dots

H. Saarikoski,1,* S. M. Reimann,2,† E. Räsänen,1 A. Harju,1 and M. J. Puska11Laboratory of Physics, Helsinki University of Technology, P.O. Box 1100, FIN-02015 HUT, Finland

2Mathematical Physics, Lund Institute of Technology, SE-22100 Lund, SwedensReceived 19 August 2004; revised manuscript received 29 October 2004; published 26 January 2005d

We study the stability and structure of vortices emerging in two-dimensional quantum dots in high magneticfields. Our results obtained with exact diagonalization and density-functional calculations show that vortexstructures can be found in various confining potentials. In nonsymmetric external potentials we find off-electron vortices that are localized giving rise to charge deficiency or holes in the electron density with rotatingcurrents around them. We discuss the role of quantum fluctuations and show that vortex formation is observ-able in the energetics of the system. Our findings suggest that vortices can be used to characterize the solutionsin high magnetic fields, giving insight into the underlying internal structure of the electronic wave function.

DOI: 10.1103/PhysRevB.71.035421 PACS numberssd: 73.21.La, 73.43.2f, 85.35.Be

I. INTRODUCTION

Vortices can occur in quantum systems which are set torotate, for example by applying an external magnetic field orby mechanical rotation.1–3 Using the Gross-Pitaevskii mean-field approach, Butts and Rokhsar3 found that in a gas ofrotating bosonic atoms which are weakly interacting by arepulsive force between them, vortices may form in a crystal-like lattice, in much analogy to patterns that emerge in rotat-ing superfluid helium. These vortex solutions appear as holessvortex linesd in the particle densities, where each single zeroof the Gross-Pitaevskii wave function corresponds to a unitvortex. With increasing angular momentum, the bosoniccloud develops a flat shape, with more and more vorticespenetrating it.

The analysis of the electronic structure of two-dimensional parabolic quantum dots4,5 sQDd in high mag-netic fields has recently shown that vortices can appear alsoin fermion systems showing many similarities to the bosoncase.6,7 High-field solutions of the mean-field density-functional theory revealed zeros in the electron densities,with electron currents circulating around them, which wereinterpreted as vortex clusters.6 Many-body techniques coulduncover vortex formation at high magnetic fields6,8 or atlarge rotation,7 giving credence to this interpretation. Withincreasing magnetic field or rotation, successive transitionsbetween stable vortex configurations were found.6

Even though emergence of vortices inside QD’s has beenpredicted by both mean-field and exact methods the interpre-tation of the effects of vortices needs careful analysis usingboth methods. As it is characteristic for mean-field theories,the self-consistent solutions of the Kohn-Sham equations forfermions, as well as the solutions of the Gross-Pitaevskiiequation, being their bosonic equivalent, break the symmetryof the quantum state. For Bose-Einstein condensates underrotation, the Gross-Pitaevskii mean-field results were shownto emerge as the correct leading-order approximation to ex-act calculations.9 For Coulomb-interacting fermions, thedensity-functional approach suffers from the relatively crudeapproximations for the exchange-correlation energy, as wellas from the problems in using a single-configuration wavefunction.10 The mean-field approach is, however, indispens-

able since analytic solutions are out of reach forNù3, and anumerically accurate direct diagonalization is also typicallyrestricted to fairly small particle numbers.

The exact diagonalization studies of the vortices in fermi-onic systems have so far concentrated on parabolically con-fined QD’s. In this case the rotational symmetry of theHamiltonian implies rotational symmetry for the particledensity in the laboratory frame of reference. Therefore, tostudy the appearance of vortices in those systems, a rotatingframe,11 conditional wave functions,6 or correlation functionshave to be examined. In contrast to the bosonic case,12 thepair correlations are not very informative for fermions due tothe disturbing influence of the exchange hole. Instead, thevortex solutions have been pinpointed either in a perturbativeapproach,7 or using a conditional wave function which fixesN−1 electrons to their most probable positions, and the wavefunction of theNth electron is calculated.6 The vortices inconditional wave functions can be seen as zeros associatedwith a phase change of integer multiple of 2p for each pathenclosing a zero. This method is not unproblematic either,since the vortices are not independent of the electron dynam-ics. The vortex locations in the conditional wave functiondepend on the positions of the fixed electron coordinates aswell as on the choice of the probing electron.

The vortices in QD’s have much in common with those inthe fractional quantum Hall effectsFQHEd. In FQHE thesystem can be approximated by Laughlin wave functionswhich attach additional vortex zeros at each electron.13 Bothin finite QD’s and infinite FQHE state the vortices carrymagnetic flux quanta and they cause strong electron-electroncorrelations. Electronic structure calculations showed that inquantum dots6 and quantum dot molecules14 vortices werefound to be bound on the electron positions, similarly to theLaughlin wave functions. A single vortex bound to an elec-tron is Pauli vortex, because it is mandated by the exclusionprinciple. In general, the number of vortices on top of elec-tron must be odd for fermions to have correct particle statis-tics. The calculations in QD’s showed also another mecha-nism to cause strong electron-electron correlations: solutionswere found with additional vortices moving between theelectrons.6 These off-electron vortices give rise to rotating

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currents of charge and a charge deficiency down to zero elec-tron density at the vortex centra.

In the FQHE the off-particle zeros are usually contrastedto on-particle zeros.15 The on-particle zeros are independentof electron coordinates except the coordinate of the electronto which the vortex is attached. On the other hand, the off-particle zeros of a given particle are not necessarily off-particle zeros of other particles. Grahamet al.16 used this factto conclude that off-particle zeros are not vortices in the realsense and therefore no charge deficiency is necessarily asso-ciated with them. Analogously to the FQHE case, the condi-tional wave functions indicate that the electrons in quantumdots see different positions for the off-particle vortices. Inour calculations, however, we find evidence that this is amanifestation of zero-point motion of vortices which causesquantum fluctuations. Our results indicate formation ofstable vortex structures where fluctuations slightly delocalizethe vortices but, nevertheless, they can be clearly seen in theexact particle and current densities as rotating current ofcharge around density minima.

In this paper we show that in quantum dots only the off-electron zeros give rise to vortex structures in the electronand current densities. We do this by applying nonsymmetricconfining potentials which leads to the localization of thevortices. In the rotationally symmetric potential vortices arenot localized and they move as the electron coordinates arechanged averaging out the effect of vortices on the particleand current densities. In rotationallynonsymmetricpoten-tials, however, the lower symmetry should causesat leastpartiald localization of vortices directly in the particle den-sity. The breaking of the circular symmetry was used alreadyby Manninenet al. in the study of Wigner localization ofelectrons in elliptical QD’s using exact diagonalizationsEDdtechniques.17 Here we show that the same trick can be ap-plied to vortices which become directly visible in the exactparticle and current densities. We find that the Pauli vorticesat the electron positions do not contribute to this effect. Inour calculations even a small asymmetry in the confiningpotential is sufficient to cause partial localization of the ad-ditional vortices. These vortices can be treated as holelikequasiparticles which have zero-point motion and thereforethey cannot be completely localized to a particular point inspace.18 We compare the exact results to the mean field so-lutions and suggest that the results can be generalized toarbitrary geometries. We conclude that the vortex structuresare stable and they can be used to classify the internal struc-ture of the many-electron wave function.

II. SYSTEM CHARACTERISTICS

We considerN electrons trapped by a confining potentialVc and subject to a perpendicular, homogeneous magneticfield B=s0,0,Bd. The system is described by anseffective-massd Hamiltonian

H = oi=1

N F spi + eAd2

2m* + Vcsr idG +e2

4peoi, j

1

ur i − r ju, s1d

whereA is the vector potential of the magnetic fieldB, m*

the effective electron mass, ande is the dielectric constant of

the medium. We apply the typical material parameters forGaAs, namely,m* /me=0.067 ande /e0=12.4. We give theenergies and lengths in effective atomic units, i.e., in Ha*

.11.86 meV and inaB* .9.79 nm.

At high magnetic fields, after complete polarization of theQD, the exchange energy results in the formation of a stableand compact structure, the so called maximum density drop-let sMDDd.19 It is a finite-size precursor of then=1 quantumHall state which assigns one Pauli vortex at each electronposition. The MDD state can be found in various QD geom-etries: in circularly confined QD’s the electrons occupy suc-cessive angular momentum states on the lowest Landaulevel, which in the parabolic case leads to a relatively flatelectron density. In rotationally nonsymmetric potentialwells, instead, the MDD window can be deduced from thekinks in the chemical potentials. In addition, the magneticfield for the MDD formation can be accurately predictedfrom the number of flux quanta penetrating the QD.20

As the magnetic field is increased, the compact electrondroplet is squeezed and eventually the repulsive interactionsbetween the fermions cause the MDD to reconstruct. Forparabolically-confined QD’s, different scenarios of the re-construction have been suggested. Chamon and Wen21 founda “stripe phase” where a lump of electrons separates from theMDD at a distance<2,B, where,B=Î" /eB is the magneticlength. Goldmann and Renn introducted projected necklacestates which they found to be lower in energy than the statesfound by Chamon and Wen.22 Geometrically unrestrictedHartree-Fock23 and CSDFTsRefs. 24 and 25d studies sug-gested that such edge reconstruction would occur with amodulated charge density wave along the edge. For a suffi-ciently small Zeeman gap, this polarized reconstruction maybe preempted by edge spin textures.26,27

The ED shows instability with respect to addition of in-ternal holes as discussed by Yang and MacDonald.28 Theseholes were recently reinterpreted as vortices in mean-fielddensity-functional calculations.6 They were found also fur-ther away from the dot center where the electron density islow. Vortices behave often like classicalslocalizedd particlesin the mean-field approach and the vortices appear as rotat-ing currents of charge with a zero in the particle density atthe vortex centra. Several charge-density-wave states thatmix different eigenstates can also be interpreted as solutionsdescribing a transport of a vortex to the center of a QD.25

Formation of vortices causes usually a broken symmetry inthe mean-field particle density, even when the Hamiltonian isazimuthally symmetric.

Tavernieret al.8 studied distribution of zeros in the exactmany-body wave function of systems containing up to 4electrons. They compared the results with the rotating-electron-moleculesREMd wave functions.29 In the regimewhere the effect of the external confining potential can beneglected, the rotating electron molecule model can providean intuitive description of the Wigner-localized electrons.Tavernier and co-workers8 found out, however, that the REMmodel is unable to predict the clustering of vortices nearelectrons.

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III. NUMERICAL RESULTS

A. Mean-field description of vortex solutions

In order to solve the many-body Schrödinger equationcorresponding to the Hamiltonians1d, we first work in themean-field picture and apply the spin-density-functionaltheorysSDFTd. For the self-consistent solution of the Kohn-Sham equations we employ a real-space scheme,30 where theexternal confining potentialVc can be arbitrarily chosenwithout symmetry restrictions. The exchange-correlation ef-fects are taken into account using the local spin-density ap-proximationsLSDAd.31 At high magnetic fields, the effect ofcurrents in the exchange-correlation potentials becomes non-negligible, and the current-spin-density-functional theorysCSDFTdsRef. 32d gives a slightly better approximation tothe ground state energy.33 The CSDFT is computationallymore demanding than the SDFT but according to our testcalculations qualitatively similar vortex structures werefound to emerge in both formalisms. We apply the SDFTthroughout this paper, since we found in these tests that itcaptures all the essential physics of these systems at muchlower computational work.

The confinement is chosen to be a two-dimensional har-monic oscillator potential with elliptic deformation, definedas

Vcsr d = 12"v0

2Sdx2 +1

dy2D , s2d

wherex and y are the major axes of the ellipse,"v is theconfinement strength, andd is the eccentricity. For compari-son, we also apply a rectangular hard-wall confinement, de-fined in Ref. 20. The parameter describing the deformationof the confining potential in this case is the side-length ratiob=Lx/Ly.

In the post-MDD domain, the SDFT predicts the forma-tion of vortices inside the QD’s. This is visualized in Fig. 1showing solutions containing up to three vortices in ellipticsa–dd and rectangularse–hd six-electron QD’s. The eccentric-ity d and the side-length ratiob have been set to 2 in thesesystems, respectively. As the vortices repel each other, theMDD states shown in Figs. 1sad and 1sed reconstruct intostates that enclose a linear vortex pattern along the longestmajor axis. There is a remarkable qualitative similarity in thehigh magnetic field behavior of these systems. However, alinear vortex cluster requires a rather large eccentricitysside-length ratiod of the QD. In the elliptic case with a confine-ment strength of"v=0.5 Ha*, the triple-vortex configurationchanges from triangular to linear whend is increased toabout 1.4.

Figure 1 shows currents induced by the magnetic field.The current is flowing clockwise around the vortices andanticlockwise on the edges of the dot. The reversal of thecurrent near the vortex core is due to inner circulation of theelectrons.38 The number of vortices in the QD increases withthe magnetic field, and the current loops of the vortices startto overlap. This causes formation of giant current loopswhich comprise several vortices.

Figure 2 shows the chemical potentials,msNd=EsNd−EsN−1d, of a rectangular sb=2d QD containing N

=7, . . . ,16 electrons. Interestingly, the regime beyond theMDD is characterized by periodic oscillations inm as a func-tion of the magnetic fieldB. The peak positions in the oscil-lations match with the transitions between adjacent vortexstates and mark the emergence of additional vortices one-by-one in the QD, presented forN=6 in Figs. 1sfd–1shd. Theoscillations get stronger as the number of electrons increases.The origin of the oscillations lies in the large reduction of the

FIG. 1. Spin-density-functional-theorysSDFTd electron densi-ties and currents in an ellipticsa–dd and rectangularse–hd six-electron quantum dot in different magnetic fields. The potential pa-rameters in effective a.u. ares"v ,dd=s0.5,2d and sLx,Lyd=s2Î2p ,Î2pd in elliptic and rectangular dots, respectively. Theincreasing of the magnetic field leads to a formation of a vortexpattern beyond the maximum density dropletsMDDd solution sB=12 Td in both geometries.

FIG. 2. Chemical potentials forN-electron rectangularsb=2dquantum dots as a function of the magnetic field. The dot is ex-pected to be fully spin-polarized. After the MDD window there areoscillations in the chemical potential which matches with the ap-pearing of additional vortices one-by-one into the quantum dot.

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Coulomb energy in connection with the vortex formation andthe coexistent pronounced localization of the electrons. Theoscillations are also visible in the total magnetization,M =−]Etot/]B, as shown in Fig. 4 in Ref. 20. The magnetizationin rectangular and elliptic six-electron QD’s are compared inFig. 3.

In elliptic QD’s the oscillations are weaker and less regu-lar than in their rectangular counterparts where the size of thedot is a constant. The soft-wall confinement makes the dotmore flexible in minimizing the total energy. These resultsare in accord with the exact diagonalization results by Gold-mann and Renn,22 who calculated the chemical potentials ofQD’s in parabolic and nonparabolic “coffee-cup” shapedconfinements. The latter confinement has a hard wall simi-larly to our rectangular potential well. Goldmann and Rennfound kinks in the chemical potentials and the kink sizesincrease with the electron number. Moreover, the kinks in thenonparabolically confined systems were found to be muchlarger than in the parabolically confined systems.

The results above suggest that the vortex formation is aconsiderable energetic effect that could be detected in appro-priate experiments. Oosterkampet al.34 measured the Cou-lomb oscillations peaksschemical potentialsd in transport ex-periments for vertical QD’s and observed additional phasetransitions beyond the MDD. They found oscillations inchemical potentials and the amplitude of these oscillationsincreased with the electron number. This data is consistentwith our calculations and could indicate vortex formation inquantum dots. A direct interpretation of their result is, how-ever, difficult due to the unknown shape of the QD sampleand its eventual sensitivity to disturbance in the experiment.Moreover, the experimental oscillations may also indicateother phenomena, such as the formation of a spin texture, forexample.26

Conditional wave functions can be introduced not only forthe analysis of the exact many-body wave function, as de-scribed in Ref. 6, but they are also useful to analyze theSDFT results. We use an auxiliary single-determinant func-tion of the Kohn-Sham orbitals which emulates the exactconditional wave function.25 This allows a study of theSDFT solutions where the electron density may have severalminima, but the vortices are not directly localized to fixed

positions. This may be due to the mixing of several eigen-states as shown in Ref. 25. For instance, the electron densi-ties for the SDFT states at 18 T and 21 T show no densityzeros. However, the corresponding conditional single-determinant functions show vortices near the fixed electronring ssee Fig. 4d.

In this picture Pauli vortices can be seen on the fixedelectrons, and additional vortices are found between the elec-trons.

The electron densities of the ellipticald=2 dot in Fig. 4show localization of both electrons and vortices. The elec-tron localization results in six density maxima which is con-sistent with earlier SDFT calculations of elliptically confinedQD’s.17 The additional vortices give rise to charge deficiencyin the electron density as seen in the right panel of Fig. 4.According to the SDFT results, the intensity of the vortexlocalization strongly depends on the magnetic field. In theabove example, the localization of vortices is partial withpronounced minima in the electron density in solutions of upto two vortices. As the magnetic field is increased further, thelocalization becomes complete i.e. the electron density van-ishes at the vortex core. This reflects the decrease of themagnetic length,B.

The results indicate that only the additional vortices be-tween the electrons have a charge deficiency associated withthem. Conditional wave functions6 and the total electron den-

FIG. 3. Magnetization of a rectangularssolid lined and ellipticalsdashed lined six-electron quantum dot as a function of the magneticfield. The ellipse eccentricityd and the rectangle side-length ratiobare both set to 2. The numbers in the figure denote the correpondingnumber of vortex holes in the electronic structure.

FIG. 4. SDFT solutions of six-electron, ellipticallysd=2d con-fined quantum dots at different magnetic fields. The confinementstrength is set to"v=0.5 Ha*. Left panel: Conditional single-determinant wave functions of the Kohn-Sham states. The fixedelectrons are marked with crosses and the probing electron is therightmost electron at the top. The contours show the logarithmicelectron density of the probe electron and the grey-scale show thephase of the wave function. The phase changes fromp to −p at thelines where shadowing changes from the darkest grey to white. Thevortices that cause charge deficiency in the center of the dot aremarked with + signs. Right panel: Electron densities show vortexholes and partial localization of the electrons forming six densitymaxima.

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sities calculated with the SDFT show that there are vorticesalso further away from the dot center. In this region the elec-tron density is usually a tiny fraction of the maximum elec-tron density. The quantum fluctuations smooth out the effectof these external vortices and they cannot be directly ob-served in the exact particle density, even though the rota-tional symmetry is broken.

B. Exact diagonalization for an elliptic dot

We compare now the above results obtained within SDFTto those of a direct numerical diagonalization of the many-body Hamiltonian matrix. We hereby focus on small particlenumbers and magnetic fields high enough such that the de-scription is mostly restricted to what in the isotropic casewould correspond to the lowest Landau level. In order todisplay the internal structure of the many-body wave func-tion, we break the spherical symmetry of the dot by applyingthe elliptical confining potentialfEq. s2dg. We assume fullpolarization of the electron droplet, and neglect the Zeemanenergy.

For the deformed case,dÞ1, where the total angular mo-mentum is not any more a good quantum numberswhile,however, we still have good parityd, the most appropriate andefficient basis set spanning the Fock space is formed by theeigenstates to the single-particle part of the Hamiltonians1d.These must be calculated numerically. We determine theMlowest ones by directly diagonalizing the single-particle partof the Hamiltonian in a basis consisting of a sufficient num-ber of corresponding Fock-Darwin35 states atd=1, which areknown analytically. Once the single-particle basis is at dis-posal, the Fock states are generated by sampling over allpossibilities to setN particles on theseM states. From thissampling, only those Fock states with defined parity and con-figuration energy36 less than a defined cut-off energy, arechosen for diagonalization. The cut-off energy was adjustedto restrict the number of Fock statessi.e., the matrix dimen-siond to be less than about 50 000. We limit the single-particle basis dimension toMø44 and use numerical inte-gration for calculating the Coulomb matrix elements. Themany-body Hamiltonians1d is then diagonalized in the ob-tained subspace. Densitiesknsr dl and real currentskj sr dl arefinally calculated in order to compare the broken-symmetrysolutions of the ED directly to the mean-field results. Here,

nsr d = oi

dsr − r id s3d

is the density operator. The real current is obtained by takingthe expectation value of

j sr d = j psr d +e

m* Asr dnsr d, s4d

where

j psr d = oi

− i"

2m* fdsr − r id¹i + ¹idsr − r idg s5d

is the paramagnetic current operator.We should note at this point that for obtaining an accurate

description of the total energy of the system, using only 44

lowest single-particle states is not sufficient for a full con-vergence of the total energy. However, the relative energydifferences and the geometrical structure of the electron andcurrent densities of the ground-state and lowest-lying stateswere converged within this basis set. Since our aim is thecomparison of the broken-symmetry many-body with themean-field solutions, rather than a detailed discussion of en-ergy spectra and excitation energies, this truncation appearedreasonable.

Figure 5 shows the electron and current densities,knsr dland kj sr dl, of the ground state and first excited state for anelliptic dot with eccentricity d=1.1 and magnetic field13.4 T. Both states have parityp=−1, and the first excitedstate is separated from the ground state by only 7.7 mHa*.The ground state shows a vortex pattern around two minimain the density and its total angular momentum isL=−25.4".The leading single-particle configuration7 of the Fock statehas the formu1001111100. . .l, with amplitudeucLu2=0.4. Thefirst excited state shows a pronounced single vortex at thedot center, with the current circulating clockwise around theorigin. The hole shows the characteristic cone shape for avortex and appears nearly localized at the dot center. Theleading configuration of the Fock states isu011111100. . .lwith amplitudeucLu2=0.7. The second excited state has parityp= +1 and is separated by a gap of 53 mHa* from theground state. It has angular momentumL=−20.4" and noclear vortex structure.

Figure 6 shows the densities and density contours for thecasesd=1.1 sad andd=1.2 sbd at an increased magnetic fieldof 17 T. Both states have parityp=−1 and angular momen-tum L=−26.7" sad and L=−28.5" sbd. In both cases, thecurrent circulates around fairly pronounced minima in thecharge density, representing two vortices at the dot center.When the eccentricity increases fromd=1.1 sad to d=1.2 sbd,the electrons begin to arrange themselves in the form of aWigner molecule with counterclockwise rotation of the cur-rent around themaximaof the electron density, and clock-

FIG. 5. Electron densitiessgrey scaled and current densitiessar-rowsd of exact diagonalizationsEDd solutions for an elliptical quan-tum dot withd=1.1. The number of electrons is 6.sad The groundstate with a two-vortex structure.sbd The first excited state with asingle vortex at the origin. The right column shows the correspond-ing electron densities at the longest major axis of the ellipse. Theconfinement strength is"v=0.5 Ha* and the magnetic field isB=13.4 T.

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wise rotation around the vorticessi.e., the minima in thedensityd, as it was also observed atd=1.1. The reversal ofthe current near the vortex core is consistent with the SDFTresults.

It is important to note here that the ground state in bothcases is a state withp= +1, which forsad is 34 mHa* and forsbd is 7.6 mHa* lower in energy. No states were found inbetween these two lowest states withp= +1 andp=−1, i.e.both states shown insad and sbd appeared as first excitedstates, respectively. The fact that the energetic sequence ofthe states changes with increasing field is, however, not sur-prising: It is well known5 from the ED studies of circularlysymmetric QD’s, that the overall structure of the spectrum isindependent of the magnetic field, the role of which ismainly to tilt the spectrum so that the minimum energy is ata different statessee Manninenet al.,17 and Maksym andChakraborty37d.

The above results from the ED are compared with themean-field calculation in Fig. 6scd for d=1.2. The resultsappear fairly similar. The largest difference naturally appearsdue to the presence of quantum fluctuations which destroythe complete localization of the vortices, i.e. the exact den-sity at the vortex center still is about one-third of the maxi-mum density, while in the mean-field result the density at thevortex center is reduced to zero.

Increasing the eccentricity still further, the localization ofelectrons leads to the formation of a charge-density-wavelikecrystal. Vortices in between the classical electron positions

become even more apparent. Figure 7 shows the ED electrondensitiessgrey scaled and current densitiessarrowsd for largereccentricityd=2. The magnetic field issad 11 T, sbd 17 T,and scd 22 T, andp=−1 in all cases.sThe p= +1 states areslightly higher in energy.d The solution at 11 T shows theMDD, and vortices form at higher magnetic fields. Thesesolutions can be compared to the SDFT solutions in Fig. 1.There is again a qualitative agreement, but now the EDshows stronger localization of electrons at both ends of thedot. While now the Fock states show much mixing and aclear dominance of a few single configurations could not beobserved, the general trend is very similar to the SDFT resultshown above in Fig. 1.

IV. SUMMARY

We have found that in noncircular confinements the vor-tices can be directly seen in the exact diagonalization elec-tron densities as holes or charge deficiency around which thecurrent is circulating. The vortex structures are stable againstfluctuations and even a slight asymmetry in the confiningpotential cause the vortices to show up in the exact many-body electron density as density minima. However, due tozero-point motion of vortices there are no zeros in the elec-tron density. The spin-density-functional calculations are inaccord with these results. They predict analogous vortex for-mation also in rectangular hard-wall quantum dots suggest-ing that the results can be generalized to a wide variety ofgeometries. The chemical potentials of the rectangular dotshow features in the energetics that could be directly com-pared to the experiments.

FIG. 6. Electron densitiessgray scaled and current densitiessar-rowsd for two-vortex solutions in elliptical QD’s. The number ofelectrons is 6.sad Exact diagonalizationsEDd result atd=1.1. sbdED result atd=1.2. scd SDFT result atd=1.2. The inset shows thecorresponding electron densities at the longest major axis of theellipse. The vortices are localized in the SDFT solution, and the EDsolution shows slight delocalization due to quantum fluctuations.The confinement strength is"v=0.5 Ha* and the magnetic field isB=17 T.

FIG. 7. Electron densitiessgrey scaled and current densitiessar-rowsd of ED solutions for elliptical quantum dots withd=2. Themagnetic field issad 17 T, sbd 21 T, andscd 22 T. The inset showsthe corresponding electron densities at the longest major axis of theellipse. The number of electrons is 6 and the confinement strength is"v=0.5 Ha*.

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In the light of these results it is justified to speak of vor-tices as real quasiparticles which might have observable ef-fects in the measurements. Vortices are not independent ofthe electron dynamics, but they can be used to characterizethe solutions in high magnetic fields which gives insight ofthe underlying internal structure of the electronic wave func-tion. Prediction of vortex formation in hard-wall potentialsgives also credence to the assumption that the vortices arerobust and largely independent of the chosen geometry of thesystem. Therefore they seem to be universal features of thephysics of the two-dimensional interacting fermion systemsin strong magnetic fields above the maximum density dropletformation.

To conclude, we briefly compare these results tobosonicsystems, where vortex formation in rotating Bose-Einsteincondensates has been much discussed both theoretically andexperimentally. There are apparent similarities between the

bosonic and fermionic case. In the composite fermion modelbosons can be turned into fermionssand the other wayaroundd by attaching fictious magnetic fluxes on top of elec-trons. Vortex formation appears as a universal phenomenonof quasi-two-dimensional quantum systems suggesting asynthetic theoretical rationale behind the phenomena.39

ACKNOWLEDGMENTS

Special thanks go to Matti Koskinen for his help with theexact diagonalization code. We also thank Matti Manninen,Ben Mottelson, and Maria Toreblad for fruitful discussions.This work has been supported by the Academy of Finlandthrough the Center of Excellence Programs2000–2005d, theSwedish Foundation for Strategic ResearchsSSFd and theSwedish Research CouncilsVRd.

*Electronic address: henri.saarikoski@hut.fi†Electronic address: reimann@matfys.lth.se1P. G. de Gennes,Superconductivity of Metals and AlloyssBen-

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