Metal clusters, quantum dots and trapped
atoms – from single-particle models to
M. Manninenaand S.M.Reimannb
aNanoscience Center, Department of Physics, FIN-40014 University of Jyv¨ askyl¨ a,
bMathematical Physics, Lund Institute of Technology, SE-22100 Lund, Sweden
In this review, we discuss the electronic structure of finite quantal systems on the
nanoscale. After a few general remarks on the many-particle physics of the harmonic
oscillator – likely being the most studied example for the many-body systems of fi-
nite quantal systems, we turn to the electronic structure of metal clusters. We
discuss Jahn-Teller deformations for the so-called ’ultimate’ jellium model which
assumes a complete cancellation of the electronic charge with the ionic background.
Within this model, we are also able to understand the stable electronic shell struc-
ture of tetrahedral (3D) or triangular (2D) cluster geometries, resembling closed
shells of the harmonic confinement, but for Mg clusters being “doubly-magic” as
the electronic shells occur at precisely twice the atom numbers in the close-packed
tetrahedra. Taking a turn to the physics of quantum dot artifical atoms, we discuss
the electronic shell structure of the quasi two-dimensional, harmonically confined
electron gas. Between the clear shell closings, corresponding to the magic numbers
in 2D, Hund’s rule acts, maximizing the quantum dot spin at mid-shell. After a brief
excursion to multicomponent quantum dots and the formation of Wigner molecules,
we turn to finite quantal systems in strong magnetic fields or, equivalently, electron
droplets that are set highly rotating. Working within the lowest Landau level, we
draw the analogy between magnetic fields and rotation, commenting on the forma-
tion of the so-called maximum density droplet (MDD) and its edge reconstruction
beyond the integer quantum Hall regime. Formation and localization of vortices
beyond the MDD, as well as electron localization at extreme angular momenta, are
discussed in detail. Analogies to the bosonic case, and the systematic build-up of
the vortex lattice of a rotating Bose-Einstein condensate at high angular momenta,
are drawn. With our contribution we wish to emphasize the many analogies that
exist between metallic clusters, semiconductor artificial atoms, and cold atoms in
Preprint submitted to Elseviertoday
arXiv:cond-mat/0703292v1 [cond-mat.mes-hall] 12 Mar 2007
One common feature of most small quantum-mechanical systems is the dis-
creteness of the quantum states. In systems with high symmetry the single-
particle energy levels are degenerate, which may lead to shell structure. This
is known to happen in free atoms, but also in nuclei . Spherical metal
clusters , where the particles move in a spherically symmetric mean field,
provide another example. In semiconductor quantum dots with circular sym-
metry, shell structure was observed by conductance spectroscopy by Tarucha
et al. . (For a review, see ). The ’universality’ of shell structure bridges
these fields of physics. However, there are also fundamental differences: In
atoms and in quantum dots, the fixed external potential dominates, leading
to Hund’s rule with maximum spin at mid-shell to resolve the degeneracy of the
spherical confinement. The valence electrons in metal clusters, or the neutrons
and protons of a nucleus, however, move in a mean-field potential determined
solely by the particle dynamics. To resolve degenaracies for non-closed shells,
metal clusters and nuclei exhibit spontaneous shape deformation, while atoms
and quantum dots do not. Consequently, the often used name ’artificial atom’
is well suited for semiconductor quantum dots, but would be misleading for
free metal clusters.
Many properties of metal clusters can be calculated by using so-called ab ini-
tio electronic structure calculations and molecular dynamics. These computa-
tional results often are in very good agreement with experimental data – as,
for example, in photoemission spectroscopy. They can pin-point the detailed
ground-state geometries of particular cluster sizes . However, many overall
features can even be understood using simple models [2,6]. This also holds for
semiconductor quantum dots, where often, simple single-particle models have
been very successful .
The purpose of this (brief, and by no means complete) review is to summarize
the simple models, their advantages and limitations in describing overall prop-
erties of metal clusters and quantum dots, and to draw analogies between these
finite quantal systems to the more recently emerging field of cold (bosonic or
fermionic) atom gases in traps.
Let us begin by looking at the simple, but relevant many-particle physics
of the harmonic oscillator. These results are then applied to understand the
jellium model for metal clusters and electronic states in quantum dots. The
universality of deformation is shortly described using simple models. We finally
turn to a comparsion of quantum dots at strong magnetic fields, and weakly
interacting bosonic systems that are set rotating.
2Many-particle physics in harmonic oscillator
The harmonic oscillator confining a single-particle is solved in about all text
books of quantum mechanics. However, adding more particles immediately
makes it more challenging to describe the system theoretically, and new inter-
esting phenomena appear. The many-body Hamiltonian is then written as
where ω0is the frequency of the confining harmonic oscillator and v(r) the
interparticle two-body interaction. The position vector ri and the Laplace
in question. Sometimes we want to use the occupation number representation
imay be three-, two- or one-dimensional depending on the system
where c+and c are the normal creation and annihilation operators (as here, for
fermions), and ?iis the single-particle energy of the form ?i= ?ω0(ni+ d/2),
d being the dimension of the system. Most conveniently, one uses the single-
particle states of the confining harmonic oscillator as a basis. It is important to
note that even if the spin index appears in this formulation of the Hamiltonian
(as a summation index), here we consider only spinless interactions, i.e. the
Hamiltonian is, as obvious from Eq. (1), independent of the spin.
The perhaps most important feature of a harmonic confinement is, that the
center-of-mass motion separates from the internal motion, regardless of the
interaction between the particles. This can easily be shown for both classical
and quantum systems . As a consequence, the selection rule for the dipole
oscillations only allows the center-of-mass excitation. In the case of simple
metal clusters and quantum dots this is the plasmon resonance, with energy
?ω0, where ω0is the frequency of the harmonic confinement . In connection
with two-dimensional quantum dots [8,9,10,11,12], the effect of the separation
of the center-of-mass motion was earlier often referred to as “Kohn’s theo-
rem” . In the case of atomic nuclei, the related excitation is called the
“giant resonance”, where the proton and neutron distributions oscillate with
respect to each other .
Another important property of the harmonic confinement is the separation
of the spatial coordinates from the center-of-mass motion (or single-particle
motion). This means that the level structure in the most general case (with
different oscillation frequencies ωialong different directions) is simply
2) + ?ω2(n2+1
2) + ?ω3(n3+1
For spherically symmetric potentials, labelling the energy levels by their ra-
dial and angular momentum indices, one obtains the harmonic energy shells
given in Table 1. Including the spin degree of freedom with a factor of two,
the “magic numbers” of the harmonic oscillator in three dimensions occur
at particle numbers 2,8,20,40,70,..., and at 2,6,12,20,30,42,... in two di-
mensions. In a non-harmonic potential, the additional degeneracy of different
Shell structure of three and two-dimensional (3D and 2D) harmonic oscillators. g
is the degeneracy of the shell and N the cumulatice number of states without spin
610 2036 12
10 2040410 20
1535 705 15 30
radial states disappears, and other degeneracies occur. A famous example is
the Wood-Saxon potential,
VWS(r) = −
1 + e(R−r)/a, (4)
frequently used in nuclear physics (where the spin-orbit interaction of the
nucleons further splits the shells ). Single-particle states in this potential
are filled following the sequence 1s,1p,1d,2s,2p,1f, etc. . The Woods-Saxon
potential Eq. (4) is a good approximation for the mean-field potential in metal
clusters , with energetically dominant magic numbers at 2, 8, 20, 40, 58,
92, etc. Note, however, that the first few magic numbers, as here 2,8 and 20,
are mainly determined by the angular momentum degeneracy and are nearly
independent of the radial shape of the spherical potential.
In the 2D harmonic confinement the noninteracting electron states can be
solved analytically also in the presence of a magnetic field , the resulting
single-particle states being the so-called Fock-Darwin states [15,16,17]. Con-
sequently, the harmonic confinement has been widely utilized when studying
the quantum Hall effect in finite systems .
3Jellium model of metal clusters
Many properties of simple metals, like alkalis, alkali earths and even aluminum
can be explained as properties of the interacting, homogeneous electron gas.
The role of the ions is then merely to keep the electron gas together at its
equilibrium density. In the jellium model , inside the metal the charge of
the ions is smoothened out and replaced by a homogeneous background charge
with the same density as the electron gas. At the surface, the background
charge goes abruptly to zero. The electron density is usually described by
the density parameter rsdefined as the radius of a sphere (in units of Bohr
radius) containing one electron: 4πr3
density of the electrons. The density functional method in the local density
approximation (LDA) is ideally suited for the jellium model which naturally
has a smooth and slowly varying electron density.
s/3 = 1/n0, with n0 being the number
This approach was first used to study metal surfaces , lattice defects 
and impurities in metals . The first application to metal clusters was made
by Martins et al. , who studied the size variation of the ionization energy.
Similar work had been successful for calculating the work function of planar
surfaces of alkali metals .
In the density functional Kohn-Sham method the electrons more in an effective
eff(r) = −eφ(r) + Vσ
where φ is the total electrostatic potential of the background charge and elec-
tron density distribution, and Vxcis the exchange-correlation potential which
depends locally on the electron density and spin polarization . In the case
of a spherical jellium cluster the effective potential resembles a finite potential
well with rounded edge. It can be well approximated by the above mentioned
Woods-Saxon potential (Eq. 4). The spherical jellium model suggests that, like
in free atoms, the ionization potential is largest for the magic clusters, and at
a minimum when only one electron occupies an open shell. The experimental
results, however, show a much richer structure as a function of the cluster size,
which in alkali metals is dominated by a marked odd-even staggering . The
reason behind are shape deformations, as described in the next section.
In the spherical jellium model for metal clusters, the the background charge is
a homogeneously charged sphere of radius R =
caused by this sphere is
3√Nrs. The (external) potential
8π?0R3(3R2− r2) if r ≤ R
if r > R .
Note that the potential inside the sphere is harmonic and can be written as
is the square of the plasmon frequency of a metallic sphere. Since the electrons
only slightly spill out from the region of the harmonic potential, the plasmon is
the dominating dipole absorption mechanism for spherical jellium clusters .
Ekardt  used the spherical jellium model in connection with the time-
dependent density functional theory to study optical absorption. He found that
the anharmonicity of the background potential caused fragmentation of the
single plasmon peak to a distribution of close-lying absorption peaks. Similar
work was subsequently done by several groups, using the RPA method [6,27].
Koskinen et al.  used shell-model methods from nuclear physics to try
to solve the electronic structure and photo-absorption of the jellium clusters
beyond the mean-field approach. For up to eight electrons, they could diago-
nalize the many-electron Hamiltonian nearly exactly. Already for 20 electrons,
however, the configuration interaction method showed a much too slow con-
vergence as a function of the size of the basis set (in fact, it was shown that the
error in the correlation energy was ∝ E−3/2
sult agreed with those of the RPA calculations. For positively charged jellium
spheres, the fragmentation of the plasmon peak disappears as the confinement
of the electrons becomes harmonic [29,28].
cut−off). For eight electrons, their re-
Historically, it is interesting to note that Martins et al.  corrected the pure
jellium results by including ion pseudo-potentials via first-order perturbation
theory, in a similar fashion than Lang and Kohn  had done for metal sur-
faces. While for planar surfaces the correction had only a minor effect (in alkali
metals), it became dominating for large metal clusters, and completely dimin-
ished the effects of the electronic shell structure of the pure jellium sphere. A
similarly large effect of the pseudopotential correction was observed for large
voids in metals, shown to be due to the low-index surfaces present in spherical
systems cut from an ideal lattice . The notion of the possible importance of
the lattice potential made the theoreticians cautious in making too strong pre-
dictions of the applicability of the jellium model to real metal clusters [32,33],
until the magic numbers of alkali metal clusters were observed .
The degeneracy of the open shell clusters should lead to Hund’s rules like
in the case of free atoms. In the spherical jellium models the clusters with
open shells should have a large total spin and magnetic moment . This
was predicted prior to the success of the jellium model by Geguzin , who
studied highly symmetric cub-octahedral Na13clusters. For free clusters, how-
ever, deformation wins over Hund’s rule and removes both the degeneracy and
We conclude that the simple spherical shell structure explains well the magic
numbers in the experimental mass spectra of sodium clusters. It lies behind
the so-called super shells [37,38] observed in alkali metal clusters , as well
as the importance of the collective plasmon resonance.
The simple jellium model also accounts for some properties of noble metals.
Recently, it has been observed that even in gold clusters some features can be
explained most easily with arguments based on the jellium model .
The similarity of small nuclei and simple metal clusters is not limited to magic
numbers and to the existence of the plasmon-type giant resonance, but extends
even to the internal deformation of the system. It is clear that the smallest
clusters can be viewed as well-defined molecules with a geometry determined
by the atomic configurations. Quantum chemistry can be used to character-
ize the ground state and spectroscopic properties of clusters with only a few
atoms. For larger clusters (N > 10) the early theories assumed spheres cut
from a metal lattice , or faceted structures with shapes determined by
the Wulff polyhedra . In reality, however, the clusters exhibit geometries
very different from these ideal structures. Many metals form icosahedral clus-
ters [42,43]. Jahn-Teller deformations are important even in quite large clus-
ters, as manifested, for example, by the odd-even staggering of the ionization
In the early cluster beam experiments, the temperature of the clusters was
lowered only by evaporative cooling. The resulting cluster temperatures were
so high that the clusters were most likely liquid . The clusters showed the
electronic shell structure as well as deformation, as determined by the splitting
of the plasmon peak . In fact, the super-shell structure could only been
seen in liquid sodium clusters. Solid clusters formed icosahedral structures
which governed the abundance and ionization potential spectra .
To model cluster deformations, Clemenger  was the first to apply the Nils-
son model familiar from nuclear physics . He was able to explain qualitative
features of the abundance spectrum of sodium clusters, including the observed
odd-even staggering. A more general model, based on the Strutinsky-model of
nuclei , was developed by Reimann et al. , and applied to triaxial ge-
ometries by Yannouleas and Landmann  as well as Reimann et al. [51,52].
It could explain nearly quantitatively the stabilities and deformation of small
4.1 Ultimate jellium model
The simplest way to include deformation in the jellium model is to assume
the uniform background charge density to be a spheroid, or an ellipsoid .
The model explains qualitatively the splitting of the plasmon peak and the
size dependence of the ionization potential of alkali metal clusters. However,
the optimal deformation shape determined by the electronic structure is not
an ellipsoid, but a more generally shaped jellium background . In the ul-
timate limit, the energy is minimized, when the background density equals
the electron density – as suggested by Manninen already in 1986 . In this
so-called ’ultimate jellium model’ (UJM) , the density of the background
is not fixed, but in a large cluster adjusts itself to correspond to r2≈ 4.2 a0, a
value close to the equilibrium electron density in sodium. (Here, a0is the Bohr
radius). The ground-state densities for clusters with N = 2 to 22 electrons are
shown as constant density surfaces in Fig. 1. Clearly, the magic numbers at
small N, here for 2, 8, and 20 electrons, correspond to spherical symmetry of
the freely deformable ’ultimate jellium’ droplet. Off-shell, however, the shapes
of the clusters exhibit breaking of axial and inversion symmetries. In general,
the resulting ground-state geometries are far from ellipsoidal. Clusters which
lack inversion symmetry, are very soft against odd-multipole deformations .
Remarkably, the results obtained from the UJM for deformations are very
close to those of ab initio calculations for sodium , as shown in Fig. 2 and
Koskinen et al.  applied the UJM to determine the shape deformations
of small nuclei. Their method gave rather good agreement with experimen-
tal results, and surprisingly, nearly exactly the same geometries as for the
electron-gas jellium. H¨ akkinen et al.  studied further the idea of this ’uni-
versal deformation’ and found that in the LDA, density functional theory
predicts similar deformations for all small fermion clusters.
This shape universality can be easily understood in systems where the parti-
cles move in a mean field caused by the particles themselves. When the num-
ber of the particles is small, there is only a small number of single-particle
states which determine the shape. For example, for four particles, only the
1s and, say, 1pxstates are filled. Consequently, the shape is prolate along the
x-direction. This corresponds to the basis of the Nilsson model .
The robustness of the shape on the specific model was further studied by
Manninen et al. , who showed that deformations of the UJM are in very
good agreement with results of the ’ultimate’ tight-binding model: the H¨ uckel
Fig. 1. Constant-density surfaces of ’ultimate jellium’ clusters with up to 22 elec-
trons. After Koskinen et al., see Ref.  for details and scales).
model for clusters .
For nuclei, the simple universal model only needs two parameters, the bulk
modulus and the average binding energy per nucleon (the first term in the so-
called mass formula ), to give good quantitative approximations to the de-
formation parameters and even excitation energies of shape isomers, as shown
in Fig. 4.
4.2 Triangles and tetrahedra
The jellium model has also been aplied to quasi two-dimensional clusters, as
for example in the early studies by Kohl et al. [62,63]. A physical realiza-
tion of two-dimensional clusters could be sodium clusters on an inert surface,
or even two-dimensional electron-hole liquids in semiconductors. Reimann et
al.  analyzed systematically the UJM ground-state shapes for quasi two-
Fig. 2. Comparison of shapes of UJM clusters (left) to those of DFT-LDA molec-
ular-dynamics methods (right), for Na-6 (upper panel) and Na-14 (lower panel).
In all cases the outer surface shown corresponds to the same particle density. Blue
spheres represent the ions. From Ref. .
Fig. 3. The three radii of the anionic sodium clusters along the principal axis, plotted
vs the number of atoms in the cluster. Down-triangles, circles, and up-triangles
correspond to Rmin, Rmiddle, and Rmax, respectively. (a) Radii corresponding to
the ground-state geometry of ab initio calculations, (b) thermally averaged radii
from room-temperature simulations, (c) radii calculated within the UJM. From
dimensional sodium clusters. Contours of the self-consistent ground-state den-
sities of these two-dimensional fermion droplets are shown in Fig. 5, calculated
for a 2D layer thickness of 3.9a0. The shape systematics reveals that for elec-
Fig. 4. Left: Shape parameter a20 for fermion clusters from 5 to 40 particles cal-
culated with the UJM (black dots connected with solid line) compared to the ex-
perimental results for even-even nuclei (stars). Right: Excitation energies of linear
isomers calculated with the UJM for nuclei (open circles) and compared to the
experimental results (black dots). From Ref. .
Fig. 5. Contours of the self-consistent ground-state densities of two-dimensional
UJM clusters for electron numbers N ≤ 2 ≤ 34, calculated for a 2D layer thickness
of 3.9a0. From Ref.  (see this Ref. for details).
tron numbers 6, 12, 20, and 30 the 2D clusters have triangular shape. Initially,
this result appeared puzzling, as these shell closures correspond to those of
the circular two-dimensional harmonic oscillator, and one should thus expect
azimuthal symmetries of the ground-state densities. The explanation was, how-
ever, that in 2D, a triangular cavity has precisely these magic numbers ,
and only in the large-N limit, the increased surface tension at the corners
makes the oscillator shells more stable. In 2D, the shell closings are rather
weak, with favorable energy minima (gaps at the fermi level) appearing mainly
in the small-N limit. Given the freedom of unrestricted shape deformations, a
pronounced odd-even staggering appears in the ground-state energies, as seen
in Fig. 6. Incidentally, these shell fillings for the triangular geometries (without
Fig. 6. Ground-state energies per electron of two-dimensional clusters, as a func-
tion of cluster size N. (The kinetic energy contribution in z-direction, tz, was sub-
tracted). The inset shows the self-consistent single-particle Kohn-Sham energies for
even particle numbers. From Ref. .
spin-degeneracy) equal precisely the number of atoms forming a close-packed
triangle. In fact, the same holds in three dimensions: at small N, thetrahedral
shell structure is prefered [66,64], with magic numbers at N = 2,8,20,40,70
and 112. These numbers correspond precisely to twice the numbers of atoms in
a close-packed thetrahedral cluster geometry (see Fig. 7). One should expect
that the compact tetrahedral geometry at an electronic magic number sta-
bilizes theses clusters. However, first principles calculations have shown that
this is not generally the case. Mg10has an overall tetrahedral shape, but is not
a perfect tetrahedron . Na20, and Mg20 are not tetrahedra, but
Au20seems to be . The experimental abundance spectrum of Mg shows a
maximum at Mg35 – but so far, there is no evidence that its geometry is
Fig. 7. Left: Ground-state shape of the UJM for 112 electrons. Right: A possible
structure of a Mg35cluster with 70 valence electrons. From Ref. .
a tetrahedron like the one shown in Fig. 7. The above results suggests that
trivalent metals on an oxide or graphite surface could favor triangular shapes.
In fact, advances in the experimental realization of surface-supported planar
clusters have been recently reported by Chiu et al. . They found magic
numbers in quasi two-dimensional Ag clusters grown on Pb islands, and stud-
ied the transition from electronic to geometric shell structure.
We finally mention that high stability of tetrahedral shapes has also been
discussed in nuclear physics [72,73], predicting tetrahedral ground states for
some exotic nuclei around110Zr (see Schunck et al., ).
5 Semiconductor quantum dots
Generally speaking, a quantum dot is a system where a small number of elec-
trons are confined in small volume in all three spatial directions. It can be,
for example, a three-dimensional atomic cluster or a two-dimensional island of
electrons formed by external gates in a semiconductor heterostructure [75,4].
In this review, we shall only consider two-dimensional semiconductor quan-
tum dots. Most often they are formed from AlGaAs-GaAs layered structures,
where a low-density 2D conduction electron gas is formed in the AlGaAs layer.
The quantum dot is formed by removing the electrons outside the dot region
with external gates (lateral dot), or by etching out the material outside the
dot region (vertical dot). In both cases, the resulting confining potential is,
to a good approximation, harmonic. The underlying lattice of the semicon-
ductor material can be taken into account by using an effective mass for the
conduction electrons, and a static dielectric constant, reducing the Coulomb
The resulting generic model for a semiconductor quantum dot is a 2D harmonic
oscillator with interacting electrons. This in fact is like a 2D jellium model,
with the simplification that now the harmonic confinement has infinite range
and the center-of-mass motion separates out exactly (Kohn’s theorem ).
This means that in the ideal case (in zero magnetic field) there is only one
dipole absorption peak, as seen in experiments .
Conductance spectroscopy can be used on one single dot. The dot is weakly
connected to leads and the current is measured as a function of the gate voltage
which determines the chemical potential and thus the number of electrons in
the dot . When the electron number in the dot is large, the energy of an
additional electron can be estimated from the capacitance C of the dot, as
∆E = e2/C. The resulting conductance then shows equidistant peaks as a
function of the gate voltage.
When the number of electrons is small, the individual single electron levels in
the dot become important and their shell structure can be seen in the con-
ductance spectrum. Tarucha et al.  were the first to successfully determine
the shell structure of circular quantum dots. Their result is shown in the lower
panel of Fig. 8, where the second derivative of the total energy of the dot is
plotted as a function of the number of electrons, N. For comparison, the corre-
sponding result of the LSDA calculation for electrons in a harmonic oscillator
is included, too.
Fig. 8. Second derivative of the total energy of electrons in a quantum dot as a func-
tion of the number of electrons. The magic numbers are shown. The experimental
result is from Tarucha et al. . The upper panel shows the calculated total spin.
The density functional Kohn-Sham method for semiconductor quantum dots
usually assumes that i) the system is two-dimensional, ii) only the conduction
electrons are considered, with an effective mass m∗and their Coulomb inter-
action screened by the static dielectric function ? of the material in question,
and iii) they move in a harmonic confinement mω2
is used for the spin-dependent exchange-correlation energy, derived from the
functionals for the 2D electron gas . For details see Refs. .
0r2. A local approximation
Shell structure with main shell fillings (magic numbers) at N = 2,6,12,20
appears very clearly in the addition energy differences ∆2(N). (See Fig. 8).
Furthermore, like in free atoms, due to Hund’s rule at mid-shell the total spin
is maximal. This means that (just like for the spherical jellium model discussed
above), any half-filled shell shows as a weak ’magic’ number, with increased
stability. This is clearly seen in Fig. 8 where the second derivative of the total
energy shows maxima at N = 4 and N = 9 in addition to the clear peaks at
the filled shells, N = 2,6, and 12. Figure 8 also shows the calculated total spin
as a function of the number of electrons in the dot. The self-consistent data
appear to agree very nicely with the experimental data. However, we notice
that this agreement becomes worse with increasing N, showing very clear
deviations between theory and experiment after the third shell, i.e. around
N = 20. Another series of experimental data, was later published by the same
group in 2001. In Ref. , addition energies for 14 different quantum dot
structures, all similar to the device used in the earlier work by Tarucha et
al. , were analyzed. Strong variations in the spectra were reported, very
clearly differing from device to device and seemingly indicating that each of
these vertical quantum dots indeed has its own properties: a comparison to the
theoreticalle expected shell structures needs to be taken with care. Progress
with vertical quantum dots was achieved more recently, where few-electron
phenomena could be studied by tunneling spectroscopy through quantum dots
in nanowires [79,80].
The self-consistent electronic structure calculations for quantum dots for some
electron numbers showed internal symmetry-breaking of the spin-density ,
leading to a static “spin-density wave” (SDW). Figure 9 shows, as an ex-
ample, the intriguing ground-state spin polarization for a quantum dot with
six electrons. For not too small densities of the electron gas, i.e. rs ≤ 6a∗
this quantum dot still has a closed-shell configuration, with S = Sz = 0.
This result is obtained from SDFT. The total density obtained by the SDFT
method is circularly symmetric, with zero net polarization (S = 0). However,
the spin polarization (which equals the difference between the spin densities
n↑− n↓normalized by the total density, n↑+ n↓), in standard SDFT breaks
the azimuthal symmetry of the confinement, showing a regular spin structure.
Fig. 9 shows this very clearly for the example of a six-electron quantum dot
at rs= 4a0. Both spin-up and spin-down densities exhibit three clear bumps,
which are twisted against each other by an angle of π/3. This resembles very
much an antiferromagnet-like structure, with alternating up- and down spins,
on a ring. Such states were obtained both with the Tanatar-Ceperley  as
well as the more recent Attaccalite–Moroni–Gori-Giorgi–Bachelet (AMGB)
 functionals for exchange-correlation. As the AMGB functional depends
explicitly on the spin polarization, there is left no doubt that the SDW states
are not simply an artefact of the ad-hoc approximation to the correlation en-
ergy, which is usually interpolated following the polarization-dependance of
the exchange energy . In quasi-one-dimensional quantum rings (see sec-
Fig. 9. DFT spin densities n↑and n↓(upper panel) and total density (n↑+ n↓) as
well as (un-normalized) spin polarization (n↑+ n↓) (lower panel) for a six-electron
quantum dot at rs= 4a∗
0, shown as 3D plots and their contours. From Ref. .
tion 7.2), these SDW states become more distinctive, as shown in Fig. 10.
The existence of the non-spherical spin-densities in quantum dots was disputed
in the literature, since the spherical symmetry of the Hamiltonian dictates
spherical symmetry [85,86]. However, as well known from nuclear physics, a
meal field theory (like KS-LSDA) can lead to internal symmetry breaking. In
some cases  it reveals the internal structure which, in fact, can be very
difficult to extract from the exact wave function. We will repeatedly meet
this problem, for example when studying vortices and localization in rotating
quantum systems, see Sections 6 and 6.4 below. (For further reading on the
internal symmetry breaking, we refer to the recent review articles [4,87,88]).
The full quantum-mechanical problem of a few electrons in a 2D harmonic os-
cillator can be solved using the so-called configuration interaction (CI) tech-
nique, numerically diagonalizing a large Hamiltonian matrix. This method
Fig. 10. Quantum rings with N = 12 and N = 13 electrons, showing antiferromag-
netic spin ordering along the ring. The maximum electron density in the 12-electron
ring is at nmax= 0.157a∗
−2. From Ref. .
is often called “exact diagonalization”, although there is always an approx-
imation due to the necessary restrictions in the basis set or the number of
configurations included. Nevertheless, up to say 6 or even 10 particles (de-
pending on the confinement strength) the results can be viewed as practically
In general, the results of the exact diagonalization agree well with those ob-
tained within the LSDA. The same spins dictated by the Hund’s rule are
obtained and the total energies agree with good accuracy. Also, the electron-
density and spin-density profiles are in excellent agreement. However, the ex-
act diagonalization can reveal the existence of the internal symmetry breaking
only via the pair correlation
gσσ?(r,r?) = ?Ψ|ˆ nσ(r)ˆ nσ?(r?)|Ψ?
where Ψ is the many-particle quantum state and ˆ nσthe spin-density operator.
This pair correlation function is also called as “conditional probability”, since
it gives the probability of finding an electron with spin σ?at r?when an electron
with spin σ is located at r. As an example, we show in Fig. 11 the pair
correlations for the above discussed six-electron quantum dot, here at rs =
(up-up) correlations. Clearly, the internal structure of the exact ground state
resembles the SDFT result described above: The probability maxima appear
on a ring, with six alternating maxima of the up- and down correlations.
0. The top panel shows the (up,down)-correlations, the bottom panel the
For a more detailed discussion, we refer to the recent work by Borgh et al. 
on broken-spin-symmetry in SDFT ground states and the reliability of SDFT.
Here we only note that SDFT and CI results generally agree very well in the
Fig. 11. Pair correlations for a quantum dot with N = 6 electrons, here at rs= 3.8a∗
obatined with the CI method. The top panel shows the (up,down)-correlations, the
bottom panel the (up-up) correlations. The black dot marks the reference point,
with given spin. From Ref. .
case of singlet states, as it was examplified above by the six-electron quantum
dot. If the true ground state is a spin-multiplett, however, SDFT introduces
an artificial splitting of multiplet states which may be misleading, and even
become a real pitfall when determining ground state energies and symmetries
where CI or other exact results to compare with, are not at hand.
The exact diagonalization method has also been used to study electron local-
ization in low-density quantum dots. At extremely low densities, the homoge-
neous electron gas forms a Wigner crystal  also in the bulk. This happens
in 3D, 2D and 1D, although in 1D the true long range order is fading with
In three dimensions, the critical value at which crystallization occurs was de-
termined to be rs= 100a0by Ceperley and Alder . In two dimensions, the
transition occurs at smaller rsvalues, according to Tanatar and Ceperley 
at rs> 37a0∗. Breaking of translational invariance in 2D lowers this value to
rs≈ 7.5a0∗. Thus, in finite systems, localization may happen at even smaller
values, as discussed, for example, by Creffield et al. , Egger et al. , or
Yannouleas and Landmann .
For finite number of electrons in a quantum dot, the localized state is often
called a “Wigner molecule” . The local density approximation can not
produce properly the localized states due to the lack of exact cancellation
of the direct and exchange Coulomb interactions. The most direct notion of
electron localization can be found by using the unrestricted Hartree-Fock ap-
proximation. The (complicated) mean-field character of the approach can lead
to broken-symmetry solutions, showing the electron localization directly in the
electron density [94,93]. This method also indicates a clear “phase transition”
point (N-dependent rs) where the crystallization occurs. However, going be-
yond Hartree-Fock to the exact diagonalization results makes the situation
more complicated: There is no clear phase transition in these finite systems,
but the localization gradually becomes stronger when rsincreases .
The most studied system in this context is the two-electron quantum dot,
the so-called “quantum dot helium” , which is in some cases exactly solv-
able [97,98,99,100,101]. Zhub et al.  have shown that at low densities
(weak confinement, small ω0) the many-particle excitation spectrum can be
described with the rotation-vibration spectrum of two localized electrons. We
will return to this method in the case of rotating systems in the next section.
Fig. 12. Electron density of a four-component quantum dot for three different
strengths of the confinement frequency, corresponding to different values of the
electron density parameter at the center of the dot: rs= 2 a0(left), 6 ao(center),
and 14 a0(right). The localization in the multicomponent LDA is made possible by
the fact that the neighboring localized electrons belong to different components as
indicated by the numbers in the contour plot.
As mentioned earlier, the LDA does not support electron localization due to
the incomplete cancellation of the direct and exchange Coulomb interaction.
The introduction of spin-dependence (LSDA) increases slightly the tendency
of localization. If more internal degrees of freedom are included, as it could be
done in the case of multi-valley semiconductors, the localization of electrons is
expected to happen also in the local approximation. This possibility was stud-
ied by K¨ arkk¨ ainen et al. . Figure 12 shows the localization of 8 electrons
in a multicomponent electron system when the confinement becomes weaker.
6Rotating systems in 2D harmonic oscillator
A semiconductor quantum dot in the presence of a perpendicular magnetic
field is a finite-size realization of the quantum Hall liquid (QHL), which has
been an exciting system of study for both experimentalists and theorists. In
fact, when Laughlin  suggested his celebrated wave function for the frac-
tional quantum Hall state, he used exact diagonalization calculations for a
finite quantum dot to test his ingenious Ansatz. Since then, it has been one of
the systems used to mimic also the infinite systems in many-particle physics
The conductance measurements through a quantum dot show that as a func-
tion of the magnetic field B, the conductance has rather complicated oscil-
lations at small B-values . These are caused by successive changes in
the quantum states of the electrons, characterized by changes in the angular
momentum and spin quantum numbers. However, at a certain field range the
oscillations disappear and it is believed that the electrons form an integer QHL
(with filling factor one). At this state the electron system is fully polarized.
The density functional theory has been extended to treat 2D electrons in the
presence of magnetic fields . In the so-called current-density functional
method the exchange-correlation energy of the electrons depends, in addition
to the electron density and polarization, also on the local current density of the
electrons. Although the method can be disputed in being not uniquely defined
in all cases and its functionals are not well established [104,105], it has been
useful in understanding the general ’phase diagram’ of the conductance, and
has been successful to suggest new kinds of symmetry-broken ground states,
with localized edge states [84,107] and vortices  (see below) as prominent
The perpendicular magnetic field in the 2D harmonic confinement has two
effects: It interacts directly with the magnetic moments of the electrons causing
a Zeeman term gµBSZB, and changes the single electron kinetic energy from
p2/2m to (p−eA)2/2m. Using the symmetric gauge for the vector potential,
A = (1/2)B(−y,x,0), and the definition of the cyclotron frequency ωc =
eB/m the single-particle Hamiltonian becomes
whereˆl is the z-component of the orbital angular momentum operator. Note
that in 2D systems this is the only component, and thus, the same as the total
angular momentum. We denote the single-particle angular momentum by m
and the many-particle angular momentum by L. Clearly, the single-particle
problem is exactly solvable, as discussed already by Fock , Darwin ,
and Landau . The single-particle energies can be written as
?nm= ?ωh(2n + |m| + 1) +1
with the angular momentum m being an integer. Figure 13 shows the single-
particle states as a function of the cyclotron frequency (magnetic field). Only
the levels with |l| ≤ 7 are shown to illustrate clearly the separation of the levels
to different Landau bands at large values of the magnetic field (large ωc). The
lowest of these consists of states with n = 0 and m = 0,
increasing order of energy.
c/4 and the radial quantum number is n = 0,1,,···
Fig. 13. Single electron states in 2D harmonic oscillator in a perpendicular magnetic
field. The levels are plotted as a function of the cyclotron frequency ωc. The levels
with n = 0 in Eq. (10) are shown as thick lines.
Normally, the Land´ e factor g in the Zeeman energy is nonzero. Consequently,
in a strong magnetic field the electron system will polarize. In this case, the
non-interacting electrons fill the N lowest energy states of the lowest Landau
level (LLL). The single-particle states of the LLL are simply
ψm(r,φ) = Cmrle−r2/2?2
where Cm is the normalization constant and ?h =
oscillator length. In the theory of QHL it is customary to describe the electron
coordinates by a complex number z = x+iy, where x = rcosφ and y = rsinφ.
The ground state of polarized noninteracting electrons is a Slater determinant
formed from the N lowest single-particle states. Conveniently, it can be written
?/mωh is the effective
(in the complex plane) as
ΨMDD(z1,z2,··· ,zN) =
where the normalization is omitted. This state is called the maximum density
droplet (MDD) and is the finite-size analog of an infinite integer QHL. Note
that the state is antisymmetric and has the total angular momentum LMDD=
N(N − 1)/2.
The electron density of the MDD is constant inside the dot, as illustrated
in Fig. 14, (note that the density of a single Slater determinant is simply
n(r) =?|ψm|2). In the case of non-interacting, polarized electrons the increase
of the magnetic field does not change the structure of the system, but the MDD
becomes smaller and smaller since ?h decreases when B (or ωc) increases.
Before we turn to the much more interesting case of interacting electrons, let
us note a few facts about the excited states of non-interacting fermions in the
The only way to excite electrons (for a fixed B or ωc) in the LLL is to increase
the single-particle angular momenta m such that the total angular momentum
increases by ∆L. This gives an excitation energy of ∆E = ?ωhL. However,
the degeneracy of the state is in general large, since there are many ways
to distribute the single-particle states in the LLL so that the total angular
momentum is LMDD+ ∆L. The wave function can be written in the complex
ΨδL= P(z1,z2,··· ,zN)
where P is any homogeneous symmetric polynomial of order ∆L. The proper
antisymmetry is provided by the determinant?(zi− zj).
6.1Interacting electrons in the LLL
The electron-electrons interactions can be included at different levels of ap-
proximations. The current-density functional theory in the LSDA takes into
account the interactions on a mean-field level and allows to include the mag-
netic field as described above. Using the material parameters (m∗and ?) of
GaAs, Reimann et al.  showed, in agreement with the experiments, that
for each electron number there exists a region where the ground state is the
maximum density droplet. This droplet slightly shrinks with increasing mag-
netic field. The “phase diagram” shown in Fig. 14 demonstrates how this
region of MDD ground states becomes narrower when the number of elec-
trons in the dot increases. (Here, the average electron density in the dot
was chosen to be approximately constant, setting the confinement strength to
?ω = 4.192N−1/4meV, which corresponds to a typical value for GaAs). When
Fig. 14. “Phase diagram” for electrons in a harmonic confinement in the presence of
a magnetic field: P denotes the region of where the polarization happens, MDD
is the maximum density droplet, CW is the region of the edge reconstruction,
and L denotes the high-field region where electron localization sets in. Schematic
densities and spin configurations of the different regions are shown at the right.
The two figures on top show calculated electron densities for 42 electrons in the
region of the MDD (left) and CW (right). The confinement strength was set to
?ω = 4.192N−1/4meV, corresponding to typical GaAs values.
the magnetic field becomes too large, the MDD breaks down. At large elec-
tron numbers, this begins from the surface of the droplet. A ring of electrons
separates from the inner, still compact, droplet. The results current-density
functional calculations suggest that in this split-off ring, the electrons are lo-
calized , as shown in Fig. 14. Again, this broken internal symmetry was
disputed in the literature. However, calculations based on other many-particle
methods have shown similar localization tendency of this so-called Chamon-
Wen edge in the correlation functions [109,110].
Fig. 15 compares the correlation functions obtained from the CI calculations,
Fig. 15. Comparison of CI correlation functions (upper panel) and mean-field den-
sities, for a 20-electron quantum dot at high rotation, or equivalently, strong mag-
netic fields. The CI results were obtained for rotation in the Lowest Landau Level
(LLL) only, for fixed angular momentum as specified. The mean-field result (lower
panel, left) was calculated in CSDFT, at an effective magnetic field of B = 3.0T
(rs= 2a0∗). The two plots at the right-hand side of the lower panel compare the
occupancies of the single-particle levels in the LLL, characterized by their single–
particle angular momentum m.
to the corresponding result in mean-field current spin density functional the-
ory (CSDFT). As an example, we here chose the 20-electron quantum dot at
high rotation, or equivalently, strong magnetic fields. The broken-symmetry
along the so-called Chamon-Wen edge is reproduced in the CI correlations.
The occupancies of the single-particle levels in the LLL, characterized by
their single-particle angular momentum m, agree remarkably well, demon-
strating the success of CSDFT in describing the correlated electronic structure
at strong magnetic fields.
6.2Rotation versus magnetic field
A magnetic field applied to the 2D harmonic oscillator leads to the simple
Hamiltonian Eq. (9). For a fixed angular momentum l the last term of the
Hamiltonian is a constant and the solutions are the harmonic oscillator ener-
gies and wave functions for the effective confinement ωh=
is an important notion: we can equivalently study the rotational spectrum of
the harmonic oscillator. For simplicity, we will now neglect the Zeeman ef-
ω0+ ωc/4. This
fect, i.e. the direct interaction between the electron spins and the magnetic
field, gµBSzB. (In fact, in semiconductors the effective Land´ e factor g can be
reduced to zero).
Similarly, for the many-particle system, even when the interactions are in-
cluded, the effect of the magnetic field for a fixed L is to increase the strength
of the confinement. Clearly, the Hamiltonian can be written as
v(|ri− rj|) +1
where nowˆL is the total angular momentum. Again, if the total angular
momentum is fixed, the last term reduces to a constant: in the case of a 2D
harmonic confinement, the effect of the magnetic field is only to put the system
in rotation, and to increase the strength of the confinement. In Fig. 16 the
results of exact diagonalization for six electrons are shown for three different
strengths of the field. Clearly, the relative structure of the spectra is very
similar, and the effect of the field is only to tilt the spectrum towards higher
angular momenta and to determine the energy scale via ωh. The rotational
spectrum alone reveals all the effects the magnetic field can have (apart from
the simple Zeeman term), making the direct comparison to other rotating
systems (like for example, cold, atomic quantum gases) meaningful.
6.3 Localization of particles at high angular momenta
We will now study the interacting system in a rotational state with a very
high angular momentum. First, let us consider fermions. For small particle
numbers the exact diagonalization technique can be used with the harmonic
oscillator states as the single-particle basis. When the angular momentum is
large all the low energy states are in the lowest Landau level (LLL) as shown
in Fig. 13, and the basis set can thus be restricted to include only the LLL.
With this restriction the matrix size will be finite (for a fixed L) and for a
small particle number no other approximations are needed.
Using the formalism of second quantization, the Hamiltonian for the polarized
electrons (we drop the spin index) is
For a fixed angular momentum L the diagonal term of the Hamiltonian gives
the energy ?ω0L for all configurations, thus, just adding a constant. The diag-
onalization of the Hamiltonian is thus reduced to the non-diagonal interaction
term. The effect of the confinement frequency ω0 (or ωh) is to provide the
15 2025 3035
Angular Momentum L
Fig. 16. Many-particle energy spectrum as a function of the total angular momen-
tum for three different values of the magnetic field (given in atomic units). From
single-particle basis and to determine the energy scale through the interaction
matrix elements vi1,i2,i3,i4. The many-particle states are completely indepen-
dent of the confinement strength, when only the LLL is included in the basis.
When studying the rotational energy spectrum, it is thus customary to plot
the interaction energy, instead of the total energy. When the angular mo-
mentum of the system increases, the systems expands and the interparticle
interactions decrease. The interaction energy then decreases with increasing
angular momentum, as seen in the figures below.
Figure 17 shows the energy spectrum calculated for four electrons as a func-
tion of the total angular momentum. Two features are distinct. First, each
appearing new energy is repeated for all higher angular momentum values.
This is due to the center-of-mass excitations. As discussed in Section 2, the
center-of-mass motion separates from the internal motion, and its excitation
energy is ?ω0n. In the LLL, each center-of-mass excitation increases the an-
gular momentum by ∆L = 1, but since this does not change the interaction
energy, it remains constant.
5 10 15 20 25 30
angular momentum L
Fig. 17. Many-particle energy spectrum (the interaction energy) for 4 electrons in a
harmonic confinement as a function of the angular momentum. The lowest energy
states are connected with a line to illustrateal the period of four.
The second important feature of Fig. 17 is the periodic oscillation of the low-
est energy state as a function of the angular momentum. These ’yrast’ states,
i.e. the states with highest possible angular momentum at a fixed energy, are
connected with a continuous line in the Figure. (Actually, the name ’yrast’
comes from Swedish language for “the most dizzy”, and originates from nu-
clear physics [112,1]. The periodic oscillation, which becomes more distinct
when the angular momentum increases, is a caused by localization of the
electrons [113,114]. Assuming that the electrons are localized in a Wigner
molecule, which in the case of four electrons has the geometry of a square, the
rigid rotation of this molecule can be quantized. The symmetry requirements
of the total wave function allow only every fourth angular momentum for a
rigid rotation . These L-values correspond precisely to the low-energy
cusps of the yrast line. The points in between can not be pure rigid rotations
and must be other internal excitations. One possibility are, for example, the
vibrational modes of the Wigner molecule.
To understand the rotation-vibration spectrum of the Wigner molecules, one
can use methods familiar from molecular physics. The corresponding energy
where I =
frequencies, and the last term gives the energy of the center-of-mass motion.
The difference between the Wigner molecule and a normal molecule is that
in the former case the Coriolis force is essential for determining the vibra-
tional frequencies. In practice, they have to be determined in a rotational
frame [113,115]. Another important difference is the drastic expansion of the
Wigner molecule as a function of the angular momentum. This causes not only
the decrease of the vibration frequency, but also the increase of the moment
2) + ?ω0(no+ 1)(16)
?mriis the moment of inertia of the molecule, ωkthe vibration
For four electrons, the vibrational modes can be solved analytically  and
the resulting energy spectrum can be constructed by considering which combi-
nations of vibrational modes and rotational states can be used to construct an
antisymmetric state. This can be done with the help of group theory [113,116].
Figure 17 shows part of the rotational spectrum, computed with exact diago-
nalization of the quantum mechanical system. It is compared to the spectrum
obtained from Eq. (16), i.e. using classical mechanics and group theory. The
figure shows an excellent agreement between the spectra. This demonstrates
clearly that at such high angular momenta, the four-electron system is just a
vibrating Wigner molecule of localized electrons.
Fig. 18. Many-particle energy spectrum (the interaction energy) for 4 electrons in a
harmonic confinement as a function of the angular momentum. Solid points: Exact
diagonalization; Squares: Model Hamiltonian (the numbers indicate the vibrational
state in question). The lower panel shows the pair correlation functions for some
The pair correlation functions shown in the lower panel of Fig. 18 further
support this conclusion. For the cusp states (as here at L = 50,54,58 for
N = 4), the pair correlations show clearly the localization of the electrons in
a square geometry, while the points in between these angular momenta reflect
the properties of the two different vibrational states.
Finally, let us discuss how this relates to the fractional quantum Hall effect.
Laughlin  showed already in his pioneering work that the maximum am-
plitude of the many-electron state in the fractional QHL
Ψq(z1,z2,··· ,zN) =
is obtained when the electrons are localized at their classical equilibrium po-
sitions. In the wave funtion above, q is an odd integer (the filling fraction of
the LLL is ν = 1/q). The localization becomes more pronounced when q in-
creases . In the region where the true Wigner crystal is formed, the above
wave function is not any more accurate. In small systems, however, already
in the region of ν = 1/3 (q = 3) the exact energy spectrum shows the peri-
odic oscillation of the yrast spectrum caused by the electron localization (see
Fig. 17). The spectrum in Fig. 18 is from the region ν ∼ 1/9.
The classical geometry of the localized electrons depends on their number .
Generally, the electrons tend to form concentric rings. Up to five electrons they
form a single ring, but for six electrons the ground state is a five-fold ring with
one electron at the center, as schematically shown in Fig. 19. This figure dis-
plays the classical equilibrium positions, for the example of six (upper panel)
and ten electrons (lower panel), respectively (after Bolton and R¨ ossler .
See also the discussion in Ref. ). The localization caused by the highly
rotational state is not limited to electrons in a harmonic confinement, but is
a more general phenomenon to occur for all particles with long-range interac-
tions. The reason behind is simple: At large angular momenta, the system can
be described by the classical rotations and vibrations of Eq. (16). The different
symmetry requirements can, however, select different allowed vibration modes
for fermions and bosons (at a given angular momentum).
Fig. 20 shows as an example the interaction energy as a function of the angular
momentum, for six bosons interacting by Coulomb repulsion. Subtracting a
smooth function of angular momentum (3rd order polynomial) from the yrast
line, pronounced and regular oscillations of period ∆L = 5 are visible in
the large-L limit, originating from the localization into a five-fold ring with
one boson at the center, just as for the fermion case discussed above. This
localization is confirmed by the pair correlations, shown as an inset in the
same figure. Reimann et al.  have furthermore shown, that the energy
spectra of small numbers of bosons and fermions are nearly identical at high
The Laughlin wave function, Eq. (17), is also applicable for bosons. In this
case, naturally the exponent q needs to be even. This suggests that there is a
relation between the boson and fermion wave functions. Since the boson wave
function is symmetric, a proper fermion wave function can be constructed by
multiplying the boson wave function with the determinant?(zi−zj). Indeed,
for the wave functions at large angular momenta this construction gives excel-
Fig. 19. Classical electron positions for small particle numbers, N ≤ 10, in a
parabolic well. After Bolton and R¨ ossler .
lent approximations for the fermion wave functions. The overlap between this
construction and the exact fermion wave function for four electrons at high
angular momenta is typically 99 % . Note, that this is not only true for
the rigidly rotating states, but also for states with internal vibrations.
Why does the rotational motion localize the particles in a harmonic confine-
ment? In the case of electrons with long-range Coulomb interactions one could
think that this is caused by Wigner crystallization. When the angular mo-
mentum increases, the electron cloud expands due to the centrifugal force and
eventually, Wigner crystallization sets in. However, we have already seen that
the external magnetic field increases the strength of the confinement. In fact,
the average electron density remains essentially constant when localization
Fig. 20. Interaction energy of N = 6 bosons as a function of the angular momentum.
The inset shows the yrast line with a smooth function of angular momentum (3rd
order polynomial) subtracted from the energies, in order to make the oscillations
more visible. The large-L limit is dominated by a regular oscillation with ∆L = 5.
The pair correlation functions to the left clearly demonstrate localization in Wigner
molecule geometries at high angular momenta. While at smaller L-values, the (1,5)
and the (0,6) configurations compete, at extreme angular momenta fivefold sym-
6.4 Vortices in polarized fermion systems
Vortex formation in type-II superconductors is a well-known phenomenon .
When the magnetic field increases, at the first critical field strength at a given
temperature, vortices penetrate the superconductor forming a regular trian-
gular lattice. Similarly, in rotating3He vortex formation has been observed
by optical measurements . Vortex formation in rotating systems has been
considered as a definite signature of superfluidity.
In the case of semiconductor quantum dots, vortex formation was discussed
theoretically by Saarikoski et al.  using the current-density functional for-
malism. Later this was confirmed by exact diagonalization calculations [123,124].
The vortices appeared when the magnetic field was increased beyond forma-
tion of the maximum density droplet, but at field strengths below those where
the fractional QHL occurs.
Just as for localization of the electrons, as discussed above, in mean-field the-
ory the vortices are visible directly as distinct minima in the total electron
density, with the electron current showing circulation of the vortex core, as
illustrated in Fig. 21. The vortices seem to localize a in a regular ’molecule’,
with geometries resembling those observed for the finite-size Wigner crystal-
lites discussed above.
Fig. 21. Electron density of a 24 electron quantum dot showing 14 vortices (left) and
the corresponding currents (right). Results from a current-spin-density functional
calculation by Saarikoski et al. From Ref. .
To analyze the vortex solutions gained by the exact diagonalization method,
is not an easy task . Naturally, for the exact solution of the many-body
Hamiltonian, the total density is circularly symmetric and one has to study
correlation functions – just as explained above for the case of Wigner local-
ization. Figure 22 shows the electron-electron pair correlation for 36 electrons
at a highly rotational state. Clearly, in addition to the exchange-correlation
hole around the reference electron, there are four distinct minima in the pair
correlation. These are four localized vortices – the reference electron pins their
position, making them visible.
Other ways to observe the internal symmetry breaking in the exact diagonal-
ization study are to break the circular symmetry, for example by an ellipsoidal
confinement [126,127], or by using perturbation theory . The electron den-
sity at the vortex core is zero, and the phase of the wave function changes by
2π when a coordinate is rotated around the vortex core. In the case of the
many-particle wave functions, these characteristics are difficult to use. It was
suggested by Saarikoski et al.  to determine the phase change of the many-
particle wave function by by fixing the positions of N − 1 coordinates when
the N’th coordinate is rotated around the vortices (fixing the other coordi-
nates fixes also the positions of the vortices). The phase maps created in this
way [108,123] show that in addition to the ’free’ vortices there is one vortex
attached to each electron. In the language of the QHL, each electron carries a
flux quantum (in the case of fractional QHL with filling factor ν = 1/3, each
electron carries three flux quanta). Electrons with attached flux quanta (or
vortices) are also called composite fermions .
In the polarized case, there is a simple way to understand the occurrence of
Fig. 22. Pair correlation functions calculated for 36 electrons. The upper panel shows
the electron-electron correlations for the MMD, L = 630 (left), for particles at
L = 706 showing four vortices (right), and for holes at the same angular momentum
(lower right). The lower panel shows the corresponding correlation function for a
bosonic four-vortex state at angular momentum L = 104. (Note the absence of the
exchange hole in the bosonic case.) From Ref. .
free vortices. They are holes in the otherwise filled Fermi see, i.e. holes in the
MDD, where all states up to the single-particle angular momentum LMDD=
N(N − 1)/2 are filled. When the angular momentum is increased we create
holes (missing electrons) corresponding to small angular momenta relative to
LMDD. Formally, we can can define the creation (annihilation) operator of a
hole as d+= c (d = c+) and write the Hamiltonian Eq. (15) in terms of these,
Note, that the interactions between the holes are the same as those between the
particles, but the second term means that the particles do not any longer move
in a strictly harmonic confinement. Naturally, the solution of this Hamiltonian
leads to an equivalent result as that of the original Hamiltonian, requiring the
same computational effort.
We will now show that – even within a limited single-particle space – the
holes localize to a Wigner molecule. Let us consider n holes in a system with
N electrons and restrict the single-particle basis to its minimum possible value
in the LLL, i.e. the maximum single-particle angular momentum being lmax=
N + n − 1. When the angular momentum of the electron system is Le =
LMDD+ ∆L, the angular momentum of the system of holes is Lh = (N +
n)(N+n−1)−Le= 2nN+n(n−1)−∆N. For example, for the N = 4 particle
system (cf. Fig. 22), Lh= 66 corresponds to such high angular momentum
that the quasi-particles (now holes) are well localized.
Fig. 23. Energy spectrum as function of the total angular momentum for 20 elec-
trons. A smooth function is subtracted from the total energy to show the oscillations
of the yrast line (thick line). The thin lines show the lowest energy states with 1, 2,
3, and 4 vortices.
This localization suggests that the excitation spectrum can be determined
from the “classical” rotations and vibrations, resulting in similar periodic os-
cillations as found above for localized electrons at high angular momenta. This
indeed is the case, as shown by Manninen et al. [124,125]. Figure 23 shows
the energy spectrum for 20 electrons as a function of angular momentum. The
yrast-line shows oscillations, in the beginning with period 2, followed by oscil-
lations of period 3 and then period 4 (in units of angular momentum). These
regions correspond to the formation of two, three, and four vortices, respec-
tively. This means that the main features of the many-particle spectrum at
these angular momenta are determined by the rotation-vibration spectrum of
localized vortices. In Section 6.5 we will see that similar oscillations reveal the
existence of vortices in rotating boson systems.
Like for fermions, also in the bosonic case, the structure of the bosonic wave
function can be understood in terms of Laughlin-type wave functions [123,125].
The simplest Ansatz for the single vortex at the center is the Bertsch-Papenbrock 
(zi− z0)ΨMDD, (19)
where z0 =
is a good approximation for the wave function calculated using only the
LLL. For example, for 10 electrons the overlap between these two states is
|?Ψ1v|Ψexact?|2= 0.90. In a large quantum dot, the center of mass can be ap-
proximated as fixed at the origin. Similarly, having n vortices in a ring, we
can approximate the wave function as
?zi/N is the center-of-mass coordinate. This wave function
(zj1− aeiα1) × ··· ×
j− an)ΨMDD, (20)
where k is the number of vortices, a is the distance of the vortices from the
origin and αj= 2πj/k. Clearly, the above wave function does not have a good
angular momentum. Projecting to good angular momentum means collecting
out states with a given power of a. We obtain a state
ΨkV = ak(N−K)S
which now corresponds to a good angular momentum M = MMDD+kK (here,
S symmetrizes the polynomial). The above wave function corresponds to the
most important configuration of the exact wave function: The n holes are next
to each other in consecutive angular momenta. Toreblad et al.  called this
state a “vortex-generating configuration”. (However, the wave function (21)
does not localize the vortices but rather keeps them de-localized at a distance
a from the origin).
6.5Vortices in rotating Bose systems
The observation of Bose-Einstein condensation in atomic traps once again in-
creased the interest in the many-particle physics of the harmonic potential (for
a review, see ). The experimental observation of vortex lattices in rotating
systems was a further milestone. By external fields, the trap can be made to
be three-dimensional or quasi-two-dimensional. In a highly rotational state,
the cloud of atoms forms a (quasi two-dimensional) disc, with the effective
confining potential in the rotating frame being
where ωr is the angular velocity of the rotation. For large enough ωr, only
the lowest energy state along the z-direction is occupied. (Note, that the rota-
tion velocity can not exceed ω0). We can then approximate the rotating Bose
system as particles confined in a 2D harmonic trap, and directly compare to
fermions confined in a quantum dot.
Fig. 24. Energy spectrum as function of the total angular momentum for N = 20
bosons (upper panel) and N = 40 bosons (lower panel), with Coulomb interactions
(L is the angular momentum, N the number of bosons). A smooth function is
subtracted from the total energy to show the oscillations of the yrast line (thick
The interaction between the atoms in the dilute condensate consists of in-
dividual scattering events which are described by the scattering length. The
contact interaction, being a standard model interaction for cold atom gases,
is written as v(ri− rj) = gδ(ri− rj), where g = 4πas?2/m, as being the
scattering length (for s-wave scattering). In the dilute gas, the total energy
per particle is proportional to the density. Consequently, in the local density
approximation the effective potential will also be proportional to the density.
In a Bose system at zero temperature, all particles are in the same quantum
state, and the density is simply ρ(r = |ψ(r)|2, where the single-particle wave
function ψ is the solution of the so-called Gross-Pitaevskii equation
0r2+ g|ψ(r)|2ψ(r) = ?ψ(r),(23)
which is a mean-field equation in close correspondence to the Kohn-Sham LDA
equations for the electron system . The nonlinearity of the equation makes
symmetry-breaking possible. Indeed, for a rotating Bose gas, the equation has
solutions showing vortex patterns very similar to the ones discussed above for
the fermion case [132,133]. Figure 24 shows the boson spectra, with oscillations
in the yrast line resembling to vortex structures as discussed above in the
For small numbers of bosons in a harmonic potential, the problem can be
solved exactly. In the case of weak interactions between the bosons, the ba-
sis set can be restricted to the lowest Landau level. The only difference to
the fermion system discussed above is, that now the wave function has to
be symmetric. For contact interactions, it has been shown that the Bertsch-
is exact for the state with a single vortex at the center (?0is now the oscillator
length of the pure confinement). This state has total angular momentum L =
N. Increasing the angular momentum creates more vortices. A second vortex
appears at L = 1.7N, the third at L = 2.1N and the fourth at L = 2.8N .
An approximation for the n vortices in a ring is again the vortex generating
state, Eq. (20), where now the fermion MDD is replaced by the ground state
of the BEC. But again, as for fermions, the exact solution is more complicated
and supports vortex localization in a much more effective way.
20 FERMIONS, L = 23820 BOSONS, L = 48
Fig. 25. Vortex-vortex correlation functions for three vortices in a fermion and boson
system with N = 20 particles. (The difference in total angular momentum is due to
the MDD in the fermion case, with LMDD= N(N −1)/2). The boson wave function
was first transformed to fermion Fock states, as described in the text. The arrows
shows the site of the reference vortex.
For bosons, the localization of vortices is not as easily seen in the pair corre-
lations as for fermions – mainly, because the bosonic occupancies in the Fock
states make it difficult to interpret the vortices directly as unoccupied states
or holes in the MDD (as in the fermion case), as the occupation number is
not limited. Nevertheless, for small particle numbers we can transform of the
boson wave function to its fermionic equivalent, by multiplying it with the
determined in this way, for N = 20 bosons with three vortices (L = 48). For
comparison, the corresponding fermion state (which in this case is not the
ground state) is shown as well (L = 328). The correlation functions appear
?(zi− zj). Figure 25 shows the vortex-vortex pair correlations
The above analysis showed very clearly that vortex localization occurs both in
the bosonic and the fermionic case, and is mapped out very directly by study-
ing the corresponding correlation functions. The rotational spectra confirmed
Figure 24 shows the energetically low-lying many-body energies for N = 20
and 40 bosons, respectively. As for fermions, we observe the oscillatory be-
havior of the yrast line. We saw above that the period of the oscillations
corresponds to the number of the first few vortices that localize on a ring.
Note, that on the horizontal axis we have now given the ratio L/N in order
to demonstrate that for bosons, the regions for different vortex numbers only
depend on L/N. This is not the case in fermion systems , where for larger
systems with N ≥ 14, the vortices appear closer of the surface of the MDD,
leaving its center unaffected.
Finally, we mention the possibility of vortex formation in boson and fermion
systems where the particles have an internal degree of freedom, like spin or
pseudospin. In this case, the single-component vortex patterns are still ob-
served, however, they are not any longer lowest-energy excitations. This holds
for fermions  as well as for bosons.
Concluding our discussion of vortices in harmonically confined quantum sys-
tems that are set rotating, we should emphasize that the vortex formation
gives characteristic oscillations in the yrast spectrum . The low-energy
states of the rotational spectrum are determined by the rigid rotation and
vibrational states of Wigner molecules of vortices . The vortex formation
is similar for bosons and fermions and it is nearly independent of the form of
the repulsive interparticle interaction [123,135].
7.1 1D harmonic oscillator
Let us finally discuss interacting electrons confined by a one-dimensional har-
monic oscillator, as well as a quasi-one-dimensional quantum ring. In an
anisotropic oscillator, Vext = (1/2)m(ω2
quencies ωyand ωzof two spatial directions is so large that the particles oc-
cupy only the lowest state in the perpendicular direction, the system becomes
zz2) choosing the fre-
Fig. 26. Upper figure: Spin polarization, as well as spin- and total electron densities
at the center of a quasi-1D wire with 12 electrons, calculated with the LSDA. Lower
figure: Electron densities for N = 6 and N = 12 noninteracting spinless fermions in
a 1D harmonic oscillator.
The 1D system of fermions is very different from the 2D and 3D cases. The
exchange interaction, or the Pauli exclusion principle, becomes dominating.
Since two electrons with the same spin can not be in the same place, in 1D
this means that electrons with the same spin can not pass each other. This
enhances drastically the tendency to form a spin density wave. In fact, an infi-
nite 1D electron gas is unstable against the so-called spin-Peierls transition: A
static spin density makes a spin-dependent mean-field potential (e.g. LSDA)
with a wave length of π/2kFand consequently opens a gap at the Fermi level.
(Remember that the Fermi surface consist of only two points in 1D). Fig-
ure 26 shows the result of an LSDA calculation for 12 electrons in a quasi-1D
harmonic potential, showing very clearly the resulting spin-density wave. The
total density shows 12 maxima corresponding to ’localized’ electrons forming
an anti-ferromagnetic chain. Even in 1D, the LSDA can not properly localize
Fig. 27. Single-particle (Kohn-Sham) densities and energy eigenvalues (inset) for
a linear finite wire with 12 electrons. Note that the last occupied state (i = 6) is
localized at either end of the wire. The energy gap between occupied and unoccupied
states is denoted by ∆ in the inset. From Ref. .
It is interesting to compare the self-consistent electron density to that of non-
interacting electrons, shown also in Fig. 26. The density for 12 spinless elec-
trons is quite similar to the LSDA result, while the density of 12 electrons
with spin has only 6 maxima since each single-particle state now occupies two
electrons. However, the similarity of the LSDA density to that of the noninter-
acting spinless electrons does not reach to the individual single-particle wave
functions. Figure 27 shows the densities of the single-particle wave functions
of the LSDA calculation. Interestingly, the last occupied state i = 6 is local-
ized at the end of the electron cloud. This end state is related to the surface
states in a metal surface. The existence of the periodic potential which ends
at the surface makes localized states possible . In our 1D case the periodic
potential is provided by the spin-Peierls transition and the static spin-density
7.2 Quantum rings
The observation of Aharonov-Bohm oscillations  and persistent currents 
have made quasi-1D quantum rings a playground for simple theories. In-
deed, the one-dimensionality as such gives a multitude of interesting prop-
erties [138,87]. Here we will only study the spectral properties of finite rings
since they are directly related to what we discussed earlier in connection to
rotational states in a 2D harmonic confinement.
Fig. 28. Many-particle energy spectrum of 8 non-interacting polarized electrons in a
strictly one-dimensional quantum ring (black dots) compared to the rotation-vibra-
tion spectrum of classical particles interacting with 1/r2interaction (open circles).
The dotted curve shows the yrast line of the polarized electrons. From Ref. .
In the strictly 1D case, the single-particle eigenvalues are ?l = ?2l2/2mR2,
where R is the radius of the ring and l the angular momentum eigenvalue. The
corresponding single-particle states are ψ(φ) = exp(ilφ). The total angular
momentum and energy for noninteracting particles is
Let us first consider noninteracting polarized (spinless) fermions. It is easy
to determine their energy as a function of the total angular momentum us-
ing Eqs. (25). The results are shown for N = 8 fermions in Fig.28 as black
dots. The yrast line shows a period of eight, suggesting that the electrons are
localized in an octagon, the downward cusps corresponding to purely rota-
tional states of the octagon. The black dots correspond to internal vibrations
of the Wigner molecule. The fact that noninteracting polarized electrons form
a Wigner molecule is a special property of 1D. It can be shown that particles
interacting with 1/r2interaction in a 1D ring have the same energy spectrum
as noninteracting particles (or particles interacting with an infinitely strong
interaction of delta-function type) . Figure 28 shows (as open circles) also
the classically determined energies
E = Erot+ Evib=
for the 1/r2interaction. Each vibrational level forms a rotational band. We
can see that the spectrum of noninteracting polarized fermions (black dots)
consists only of points at the classical energies. In fact, for electron with spin
and infinite strong delta-function interaction (v(r) = Aδ(r), where A → ∞)
one obtains all the classical points (open circles).
The reason why noninteracting spinless electrons localize and have vibrational
modes simply follows from the fact that the electrons can not pass each other.
If the electron-electron distance is d, each electron is then localized between
its neighbors in a region 2d. Its kinetic energy will then be proportional to
1/d2. This effectively leads to a 1/r2interaction between the electrons.
Interacting electrons in 1D systems have been extensively studied using the
Hubbard model (for reviews see [138,87]). The energy spectrum can be solved
exactly using the Bethe Ansatz . There, several analytic results exist.
For a half-filled Hubbard band (with one electron per site) it is rather easy to
show that the large U-limit the Hubbard model becomes an anti-ferromagnetic
Heisenberg model. However, the Heisenberg model seems to be a good approx-
imation also for small filling [140,87]. This is important, since the low-filling
limit of the Hubbard model approaches to free electrons with delta interac-
tion (this is the same as the tight-binding model approaching the free electron
model at the bottom of the band ). Thus, also free electrons with spin lo-
calize in anti-ferromagnetic order as long as they have strong enough repulsive
interactions between them.
Koskinen et al.  performed exact diagonalization calculations for elec-
trons confined in a quasi-1D ring described with the external 2D potential
ticles is shown in Fig. 29 for two different values of the narrowness of the ring.
The upper panel corresponds to a very narrow ring. In this case, the differ-
ent vibrational bands are clearly separated and correspond quantitatively to
the energies determined by solving the vibrational frequencies of the classical
linear chain of electrons on the ring. The lower panel shows the result ob-
tained for a wider, less one-dimensional ring. In this case, only the vibrational
ground state is clearly separated, with the different spin-states separating in
energy. With high accuracy, these different spin-states correspond to those of
an antiferromagnetic Heisenberg model for six electrons on a ring .
0(r − r0)2/2, where r = (x,y). The rotational spectrum for six par-
In narrow quantum rings the rotational spectrum is very robust. It is insensi-
tive to the interparticle interaction or the specific model for the confinement.
Even the discrete Hubbard model gives similar results as the continuum ap-
proaches . However, this demonstrates once more that the most clear in-
dication of Wigner molecules in the ground states of high-symmetry systems
can be obtained by analyzing the rotation-vibration spectrum.
N=6 r = 2 C = 4
Angular Momentum M
Angular Momentum M
N=6 r = 6 C = 25
Fig. 29. Energy spectra for two quasi-one-dimensional continuum rings with six
electrons (in zero magnetic field). The upper panel is for a narrow ring and it shows
several vibrational bands. The lower panel is for a wider ring which shows stronger
separation of energy levels corresponding to different spin states (shown as numbers
next to the energy levels). Note that also the narrow ring has the same spin-ordering
of the nearly degenerate state as expanded for the lowest L = 0 state.
In this short review, we summarized some characteristic aspects of finite quan-
tal systems, that have their origin in the quantized level structure in – and
beyond – the mesoscopic regime. We discussed shell structure and deforma-
tion, as well as the ocurrence of Hund’s rule in finite fermion systems, con-
jointly for metallic clusters, quantum dots in semiconductor heterostructures,
or cold atoms in traps – seemingly different, but nevertheless in many aspects
rather similar quantum systems. Like for atoms, shell structure does not only
determine the stability and chemical inertness of metallic clusters, but also
determines the conductance of a small quantum dot – both close to, and far
away from equilibrium.
The experimental realization of Bose-Einstein condensation in an atomic gas [143,144,145,146]
opened up a whole new research field on ultra-cold atoms and coherent mat-
ter. In a cloud of bosonic atoms that is set rotating, vortices may form. We
discussed the fact that this vortex formation is not unique for bosonic systems,
but may occur in a very similar way for (non-paired) fermions under rotation,
showing many analogies to the physics of the quantum Hall effect. Extreme
rotation causes strong correlations, and the system is formally equivalent to
charged particles in a strong magnetic field. We finally gave a short summary
of the physics of a finite fermionic system in quasi one dimension.
As a final remark, we wish to emphasize that the many analogies existing
between nanostructures such as quantum dots and quantum wires, and cold
atom gases will become more important in the future – last but not least
due to the fact that these systems can be built much more “clean”, and thus
more coherent, than their semiconductor counterparts. An example for the
cross-fertilization between these different sub-fields of physics, is the recently
discussed possibility of of van-der-Waals blockade , which is expected
to play a key role in transport experiments on confined cold atoms, and in
atomtronic devices .
This work was financially supported by the Swedish Research Council and the
Swedish Foundation for Strategic Research, as well as uthe European Com-
munity project ULTRA-1D (NMP4-CT-2003-505457). We thank J. Akola, M.
Borgh, P. Singha Deo, H. H¨ akkinen, G. Kavoulakis, M. Koskinen, P. Lipas,
B. Mottelson, P. Nikkarila, M. Toreblad, S. Viefers, and Y. Yu for their col-
laboration on the subjects discussed in this review.
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