To model the experimental quantum dot device, we take, following Ref. [41],
$\hbar\omega_{x}=4.23$ meV, $\hbar\omega_{y}=5.84$ meV, $\kappa=12.5$, and
$\gamma=0.862$. The corresponding anisotropy is $\omega_{y}/\omega_{x}=1.38$,
indicating that the quantum dot considered here is closer to being circular
than in other experimental systems [45, 80].

Figure 9: (Color online) Differentiated current $\rmd I/\rmd V_{\mathrm{pg}}$
at $V_{\mathrm{bias}}=2.5$ mV (the subscript ${\rm pg}$ denotes the plunger
gate). Gray striped regions (red online) marked by symbols correspond to
positive (peaks) $dI/dV_{\mathrm{pg}}$. The dark black region (also black
online) corresponds to negative $dI/dV_{\mathrm{pg}}$. Electron numbers $N$
are indicated. Transitions between the one-electron ground state and the 2$e$
spin-singlet ground (excited for $B>3.8$ T) state ($S_{0}$), spin-triplet
excited (ground for $B>3.8$ T) state ($T_{+}$), spin-singlet excited state
($S_{2}$), and spin-triplet plus center-of-mass excited state ($T_{+,{\rm
CM}}$) are labeled.  Figure 10: (Color online) Calculated exact-
diagonalization energy spectrum in a magnetic field, referenced to
$2\hbar\sqrt{\omega_{0}^{2}+\omega_{c}^{2}/4}$, of a 2$e$ dot with anisotropic
harmonic confinement (for the dot parameters, see text). We have adopted the
notation $(N_{x},N_{y},n,m)$, where $(N_{x},N_{y})$ refer to the CM motion
along the $x$ and $y$ axes and $(n,m)$ refer to the number of radial nodes and
angular momentum of the relative motion in the corresponding circular dot.
Inset: The EXD spectrum of the corresponding circular dot. Only the $(n,m)$
indices are shown, since $N_{x}=N_{y}=0$ for all the plotted curves. Solid
lines denote singlets. Dashed lines denote triplets.

As shown in Ref. [41], the experimental findings can be quantitatively
interpreted by comparing to the results of the EXD calculations for two
electrons in an anisotropic harmonic confinement potential with the parameters
listed above. All the states observed in the measured spectra (as a function
of the magnetic field) can be unambiguously identified [41] with calculated
ground-state and excited states of the two-electron Hamiltonian (compare
figure 9 and figure 10).

Moreover, the calculated magnetic-field-dependent energy splitting,
$J_{\rm{EXD}}(B)=E^{t}_{\rm{EXD}}(B)-E^{s}_{\rm{EXD}}(B)$, between the two
lowest singlet ($S_{0}$) and triplet ($T_{+}$) states is found to be in
remarkable agreement with the experiment [see figure 11].

Figure 11: (Color online) Comparison between the lowest-triplet/lowest-singlet
energy splitting [$J(B)$] calculated with different methods and the
experimental results (open squares). Solid line (online magenta): EXD. Dotted
line (online green): GHL. Dashed line (online red): RHF. For the parameters
used in the calculation to model the anisotropic QD, see text.

A deeper understanding of the structure of the many-body wave function can be
acquired by comparing the measured $J(B)$ with that calculated within the GHL
and RHF approximations. To facilitate the comparisons, the calculated
$J_{\rm{GHL}}(B)$ and $J_{\rm{RHF}}(B)$ curves are plotted also in figure 11,
along with the exact-diagonalization result and the measurements. Both the RHF
and GHL schemes are appealing intuitively, because they minimize the total
energy using single-particle orbitals. It is evident, however, from figure 11
that the RHF method, which assumes that both electrons occupy a common single-
particle orbital, is not able to reproduce the experimental findings. On the
contrary, the generalized Heitler-London approach, which allows the two
electrons to occupy two spatially separated orbitals, appears to be a good
approximation. Plotting the two GHL orbitals [see figure 12] for the singlet
state clearly demonstrates that the two electrons do not occupy the same
orbital, but rather fill states that are significantly spatially separated.

Figure 12: (Color online) Single-particle UHF orbitals (modulus square) that
are used in the construction of the GHL wave function in (5.3). Arrows
indicate up and down spins. For the parameters used in the calculation to
model the anisotropic QD, see text. Lengths in nm and orbital densities in
10-3 nm-2.

The UHF orbitals from which the GHL singlet state is constructed [see (5.3)]
are displayed on figure 12 for both the $B=0$ and $B=3.8$ T cases. The spatial
shrinking of these orbitals at the higher $B$-value illustrates the
“dissociation” of the electron dimer with increasing magnetic field. The
asymptotic convergence (beyond the ST point) of the energies of the singlet
and triplet states, [i.e., $J(B)\rightarrow 0$ as $B\rightarrow\infty$] is a
reflection of the dissociation of the 2$e$ molecule, since the ground-state
energy of two fully spatially separated electrons (zero overlap) does not
depend on the total spin. We stress again that the RHF, which corresponds to
the more familiar physical picture of a QD-Helium atom, fails to describe this
dissociation, because $J_{\rm{RHF}}(B)$ diverges as the value of the magnetic
field increases.

In contrast to the RHF, the GHL wave function is able to capture the
importance of correlation effects. Further insight into the inportance of
correlations in our QD device can be gained through inspection [41] of the
conditional probability distributions [see (1.1)] associated with the EXD
solutions; see an illustration in figure 13. Indeed, already at zero magnetic
field, the calculated CPDs provide further support of the physical picture of
two localized electrons forming a state resembling an H2-type [23, 46, 41]
Wigner molecule [20, 188].

Figure 13: (Color online) CPDs extracted from the exact-diagonalization wave
function for the singlet state for $B=0$ and $B=3.8$ T. The CPD expresses the
conditional probability for finding the second electron at position ${\bf r}$
given that the first electron is located at ${\bf r}_{0}$ (denoted by a heavy
solid dot). For the parameters used in the calculation to model the
anisotropic QD, see text. Lengths in nm and CPDs in arbitrary units.

####  5.1.4 Degree of entanglement.

Further connections between the strong correlations found in our microscopic
treatment and the theory of quantum computing [48] can be made through
specification of the degree of entanglement between the two localized
electrons in the molecular dimer. For two electrons, we can quantify the
degree of entanglement by calculating a well-known measure of entanglement
such as the von Neumann entropy [42, 200] for indistinguishable particles. To
this effect, one needs to bring the EXD wave function into a diagonal form
(the socalled “canonical form” [200, 201]), i.e.,

$\Psi^{s,t}_{\rm{EXD}}({\bf r}_{1},{\bf
r}_{2})=\sum_{k=1}^{M}z^{s,t}_{k}|\Phi(1;2k-1)\Phi(2;2k)\rangle,$ (5.6)

with the $\Phi(i)$’s being appropriate spin orbitals resulting from a unitary
transformation of the basis spin orbitals $\psi(j)$’s [see (5.5)]; only terms
with $z_{k}\neq 0$ contribute. The upper bound $M$ can be smaller (but not
larger) than $K$ (the dimension of the single-particle basis); $M$ is referred
to as the Slater rank. One obtains the coefficients of the canonical expansion
from the fact that the $|z_{k}|^{2}$ are eigenvalues of the hermitian matrix
${\cal A}^{\dagger}{\cal A}$ [${\cal A}$, see (5.5), is antisymmetric]. The
von Neumann entropy is given by ${\cal
S}=-\sum_{k=1}^{M}|z_{k}|^{2}\log_{2}(|z_{k}|^{2})$ with the normalization
$\sum_{k=1}^{M}|z_{k}|^{2}=1$.

Figure 14:  Von Neumann entropy for the lowest singlet EXD state of the
elliptic dot as a function of the magnetic field $B$. On the top, we show
histograms for the $|z_{k}|^{2}$ coefficients [see (5.6)] of the singlet state
at $B=3.8$ T (left) and $B=8.0$ T (right) illustrating the dominance of two
determinantal configurations (in agreement with the generalized Heitler-London
picture). Note the small third coefficient $|z_{3}|^{2}=0.081$ for $B=8.0$ T.
For the parameters used to model the experimental device, see text.

The EXD singlet has obviously a Slater rank $M>2$. The von Neumann entropy for
the EXD singlet (${\cal S}^{s}_{\rm{EXD}}$) is displayed in figure 14. It is
remarkable that ${\cal S}^{s}_{\rm{EXD}}$ increases with increasing $B$, but
remains close to unity for large $B$, although the maximum allowed
mathematical value is $\log_{2}(K)$ [for example, for $K=79$,
$\log_{2}(79)=6.3$]. The saturation of the entropy for large $B$ to a value
close to unity reflects the dominant (and roughly equal at large $B$) weight
of two configurations in the canonical expansion [see (5.6)] of the exact-
diagonalization wave function, which are related [42] to the two terms in the
canonical expansion of the GHL singlet. This is illustrated by the histograms
of the $|z_{k}^{s}|^{2}$ coefficients for $B=3.8$ T and $B=8.0$ T in figure 14
(top). Notice that the ratio $|z_{2}|^{2}/|z_{1}|^{2}$ reflects the extent of
the overlap between the two GHL orbitals [42], with the ratio increasing for
smaller overlaps (corresponding to a more complete dissociation of the Wigner
molecule).

The above discussion illustrates that microscopic calculations that are shown
to reproduce experimental spectra [41] can be used to extract valuable
information that allows assessment of the suitability of a given device for
quantum computations.

###  5.2 Two-electron circular dots at zero magnetic field

In section 2.2, we illustrated the formation of “rotating electron molecules”
in the case of a two-electron circular QD, where one needs to consider
restoration of the rotational symmetry as well, in addition to the restoration
of the total spin. There, we focused on properties of the ground state
($L=0$).

Figure 15: (Color online) The calculated spectrum of a two-electron circular
parabolic quantum dot, with $R_{W}=200$. The quantum numbers are $(N,M,n,m)$
with $N$ corresponding to the number of radial nodes in the center of mass
(CM) wavefunction, and $M$ is the CM azimuthal quantum number. The integers
$n$ and $m$ are the corresponding quantum numbers for the electrons’ relative
motion (RM) and the total energy is given by $E_{NM,nm}=E^{\rm CM}_{NM}+E^{\rm
RM}(n,|m|)$. The spectrum may be summarized by the “spectral rule” given in
the figure, with an effective rigid moment of inertia $C=0.037$ (corresponding
to an angular momentum ${\cal L}=\hbar m$), the phonon for the stretching
vibration $\hbar\omega_{s}=3.50$, and the phonon for the bending vibration
coincides with that of the CM motion, i.e.,
$\hbar\omega_{b}=\hbar\omega_{0}=2$. The quantum numbers
$(N_{0},M_{0},n_{0},m)$ specifying each rotational band are given at the
bottom, with $m=0,1,2,\ldots$ (the levels $m=0$ and $m=1$ in each band may not
be resolved on the scale of the figure). We note that the energy separation
between levels in a given rotational band increases as $(2m+1)$ with
increasing $m$, which is a behavior characteristic of a rigid rotor. All
energies are in units of $\hbar\omega_{0}/2$, where $\omega_{0}$ is the
parabolic confinement frequency.

In this section, we will further examine the excitation spectra of a two-
electron QD by using the rather simple exact solution of this problem provided
through separation of the center-of-mass and inter-electron relative-distance
degrees of freedom [50]. The spectrum obtained for $R_{W}=200$ (figure 15),
exhibits features that are characteristic of a collective rovibrational
dynamics, akin to that of a natural “near-rigid” triatomic linear molecule
with an infinitely heavy middle particle representing the center of mass of
the dot. This spectrum transforms to that of a “floppy” molecule for smaller
value of $R_{W}$ (i.e., for stronger confinements characterized by a larger
value of $\omega_{0}$, and/or for weaker inter-electron repulsion), ultimately
converging to the independent-particle picture associated with the circular
central mean-field of the QD.

Further evidence for the formation of the electron molecule and the emergence
of a rovibrational spectrum was found through examination [50] of the
conditional probability distributions for various states $(N,M,n,m)$ (see the
caption of figure 15 for the precise meaning of these quantum numbers labeling
the spectra). As an example, we display in figure 16 the CPD for the bottom
state ($m=0$) of the rotational band $(1,0,1,m)$ (not shown in figure 15); it
reveals that this state corresponds to a vibrational motion of the electron
molecule both along the interelectron axis (one excited stretching-mode
phonon; see figure 15) and perpendicularly to this axis (two excited bending-
mode phonons; see figure 15).

It is instructive to note here certain similarities between the formation of a
“two-electron molecule” in man-made quantum dots, and the collective
(rovibrational) features observed in the electronic spectrum of doubly-excited
helium atoms [202, 203, 204].

Figure 16: CPD of the excited multi-vibrational state (1,0,1,0) of a $2e$
circular parabolic QD with $R_{W}=200$ (see text for details). The solid dot
portrays the position of the reference point ${\bf r}_{0}=(d_{0},0)$, where
$d_{0}=2.6$ is half the interelectron distance at the ground state
$(0,0,0,0)$. Distances are in units of $l_{0}\sqrt{2}$; the scale of the
vertical axis is arbitrary.

###  5.3 Historical significance of the two-electron problem

In spite of being the simplest many-body system, the significance of the
problem of two interacting electrons confined in an external potential cannot
be overstated. Historically it played a central role in the development of the
quantum theory of matter through the failure of the Bohr-type semiclassical
models to account for the natural He atom. Most recently it has influenced the
development of several fields like nonlinear physics, atomic physics,
semiconductror quantum dots, and quantum computing.

It is instructive to make here a historical detour. Indeed, the failure of
Bohr-type semiclassical models, based on the orbiting of spatially correlated
(antipodal) electrons in conjunction with the Bohr-Sommerfeld quantization
rule, to yield a reasonable estimate of the ground state of the He atom
signaled a looming crisis in physics in the 1920’s, which Bohr himself, as
well as others, had been keenly aware of, as summarized succintly by
Sommerfeld: “All attempts made hitherto to solve the problem of the neutral
helium atom have proved to be unsuccessful” [205]; see also the 10th chapter
entitled “It was the Spring of hope, it was the Winter of despair” in the book
by Pais [206], the review by Van Vleck [207] and the book by Born [208].

While, since, numerical solutions of the two-electron Schrödinger equation
provided a quantitative resolution to the problem, the first successful
semiclassical treatment of the three-body Coulomb system awaited till 1980
[209, 210].

Furthermore, based on rather general group-theoretical arguments arising from
the observation of hierarchies with lower symmetry in the excited spectra, and
motivated by ideas originating in nuclear-physics spectroscopy, it has been
discovered in the late 1970’s and early 1980’s that electron correlations in
doubly excited He lead to quantization of the spectrum much like in a linear
triatomic molecule, $e$-He2+-$e$. This molecular picture, with near rigidity
and separability, results in “infinite sequences of vibrational levels, on
each of which is built an infinite sequence of rotational levels” [202, 211,
212].

The two previous section 5.1 and section 5.2 describing the formation and
properties of a 2$e$ Wigner molecule in a single QD may be viewed as the
culmination of this historical background. Interestingly, as in the
aforementioned semiclassical treatments, the collinear configuration plays a
special role in the molecule-like model, serving perhaps as “partial
vindication” of the geometry considered originally by Niels Bohr.

##  6 Rotating electron molecules in two-dimensional quantum dots under a
strong magnetic field: The case of the lowest Landau level
($\omega_{c}/2\omega_{0}\rightarrow\infty$)

###  6.1 REM analytic trial wave functions

In the last ten years, and in particular since 1999 (when it was explicitly
demonstrated [20] that Wigner crystallization for small systems is related to
symmetry breaking at the unrestricted Hartree-Fock mean-field level), the
number of publications addressing the formation and properties of Wigner (or
electron) molecules in 2D QDs and quantum dot molecules has grown steadily
[20, 21, 23, 24, 27, 50, 101, 102, 128, 132, 167, 171, 175, 188, 189, 195,
213, 214, 215, 216, 217, 218, 219, 220, 221]. A consensus has been reached
that rotating electron molecules are formed both in zero [21, 23, 24, 46, 50,
101, 167, 175, 188, 189, 195, 213, 214, 215, 216, 217, 218, 219] and high [26,
27, 46, 102, 128, 131, 132, 171, 220, 221] magnetic fields.

At $B=0$, in spite of considerable differences explored in this report (see
next paragraph), formation of REMs in quantum dots is driven by the same
physical factors as Wigner crystallization in infinite 2D media, i.e., when
the strength of the interelectron repulsion relative to the zero-point kinetic
energy ($R_{W}$) exceeds a certain critical value, electrons spontaneously
crystallize around sites forming geometric molecular structures. At high
magnetic fields, the formation of Wigner molecules may be thought of as
involving a two-step crystallization process: (I) the localization of
electrons results from the shrinkage of the orbitals due to the increasing
strength of the magnetic field; (II) then, even a weak interelectron Coulomb
repulsion is able to arrange the localized electrons according to geometric
molecular structures (thus this process is independent of the value of
$R_{W}$). It has been found [24, 26, 27, 128] that the molecular structures at
high $B$ coincide with the equilibrium configurations at $B=0$ of $N$
classical point charges [112, 113, 114, 222].

Due to the small finite number, $N$, of electrons, however, there are two
crucial differences between the REM and the bulk Wigner crystal. Namely, (I)
the crystalline structure is that of the equilibrium 2D configuration of $N$
classical point charges, and thus consists of nested polygonal rings, in
contrast to the well known hexagonal bulk crystal;

(II) in analogy with the case of 3D natural molecules, the Wigner molecules
rotate as a whole (collective rotations); they behave, however, as highly
floppy (non-rigid) rotors.

A most striking observation concerning the REMs is that their formation and
properties have been established with the help of traditional ab initio many-
body methods, i.e., exact diagonalization, [50, 128, 131, 167, 171, 213, 221]
quantum Monte Carlo [78, 188, 195, 196, 216], and the systematic controlled
hierarchy [20, 22, 23, 24, 25, 27, 102, 217] of approximations involving the
UHF and subsequent post-Hartree-Fock methods. This contrasts with the case of
the Jastrow/Laughlin [55] and composite-fermion [56, 57] wave functions, which
were constructed through “intuition-based guesswork.”

In spite of its appearance in the middle nineties and its firm foundation in
many-body theory, however, the REM picture had not, until recently,
successfully competed with the CF/JL picture; indeed many research papers
[181, 223, 224, 225, 226, 227, 228] and books [133] describe the physics of
quantum dots in high magnetic fields following exclusively notions based on
CF/JL functions, as expounded in 1983 (see Ref. [55]) and developed in detail
in 1995 in Ref. [229] and Ref. [174]. One of the main obstacles for more
frequent use of the REM picture had been the lack of analytic correlated wave
functions associated with this picture. This situation, however, changed with
the recent explicit derivation of such REM wave functions [24].

The approach used in Ref. [24] for constructing the analytic REM functions in
high $B$ consists of two-steps: First the breaking of the rotational symmetry
at the level of the single-determinantal unrestricted Hartree-Fock
approximation yields states representing electron molecules. Subsequently the
rotation of the electron molecule is described through restoration of the
circular symmetry via post Hartree-Fock methods, and in particular Projection
Techniques [18]. The restoration of symmetry goes beyond the single
determinantal mean-field description and yields multi-determinantal wave
functions.

In the zero and low-field cases, the broken symmetry UHF orbitals need to be
determined numerically, and, in addition, the restoration of the total-spin
symmetry needs to be considered for unpolarized and partially polarized cases.
The formalism and mathematical details of this procedure at $B=0$ have been
elaborated in previous sections.

In the case of high magnetic fields, the spins of the electrons are fully
polarized. Furthermore, one can specifically consider the limit when the
confining potential can be neglected compared to the confinement induced by
the magnetic field, so that the Hilbert space is restricted to the lowest
Landau level. Then, assuming a symmetric gauge, the UHF orbitals can be
represented [24, 230] by displaced Gaussian analytic functions, centered at
different positions $Z_{j}\equiv X_{j}+\rmi Y_{j}$ according to the
equilibrium configuration of $N$ classical point charges [112, 113, 114, 222]
arranged at the vertices of nested regular polygons (each Gaussian
representing a localized electron). Such displaced Gaussians in the lowest
Landau level are written as

$u(z,Z_{j})=(1/\sqrt{\pi})\exp[-|z-Z_{j}|^{2}/2]\exp[-\rmi(xY_{j}-yX_{j})],$
(6.1)

where the phase factor is due to the gauge invariance. $z\equiv x+\rmi y$, and
all lengths are in dimensionless units of ${l_{B}}\sqrt{2}$ with the magnetic
length being $l_{B}=\sqrt{\hbar c/eB}$. Note that expression (6.1) is a
special case of the more general expression (2.24) for a displaced Gaussian
which corresponds to situations with smaller magnetic fields when the
restriction to the lowest Landau level breaks down. The notation $z\equiv
x+\rmi y$ is associated with positive angular momenta for the single-particle
states in the lowest Landau level. Ref. [24] used $z\equiv x-\rmi y$ and
negative single-particle angular momenta in the lowest Landau level. The final
expressions for the trial wave functions do not depend on these choices.

Table 1: Comparison of yrast-band energies obtained from REM and EXD
calculations for $N=6$ electrons in the lowest Landau level, that is in the
limit $B\rightarrow\infty$. In this limit the external confinement can be
neglected and only the interaction energy contributes to the yrast-band
energies. Energies in units of $e^{2}/(\kappa l_{B})$. For the REM results,
the (1,5) polygonal-ring arrangement was considered. The values of the
fractional filling may be obtained for each $L$ as $\nu=N(N-1)/(2L)$.

  * $L$ | REM | EXD | Error (%) | $L$ | REM | EXD | Error (%)
---|---|---|---|---|---|---|---
70 | 2.3019 | 2.2824 | 0.85 | 140 | 1.6059 | 1.6006 | 0.33
75 | 2.2207 | 2.2018 | 0.85 | 145 | 1.5773 | 1.5724 | 0.31
80 | 2.1455 | 2.1304 | 0.71 | 150 | 1.5502 | 1.5455 | 0.30
85 | 2.0785 | 2.0651 | 0.65 | 155 | 1.5244 | 1.5200 | 0.29
90 | 2.0174 | 2.0054 | 0.60 | 160 | 1.4999 | 1.4957 | 0.28
95 | 1.9614 | 1.9506 | 0.55 | 165 | 1.4765 | 1.4726 | 0.27
100 | 1.9098 | 1.9001 | 0.51 | 170 | 1.4542 | 1.4505 | 0.26
105 | 1.8622 | 1.8533 | 0.48 | 175 | 1.4329 | 1.4293 | 0.25
110 | 1.8179 | 1.8098 | 0.45 | 180 | 1.4125 | 1.4091 | 0.24
115 | 1.7767 | 1.7692 | 0.42 | 185 | 1.3929 | 1.3897 | 0.23
120 | 1.7382 | 1.7312 | 0.40 | 190 | 1.3741 | 1.3710 | 0.23
125 | 1.7020 | 1.6956 | 0.38 | 195 | 1.3561 | 1.3531 | 0.22
130 | 1.6681 | 1.6621 | 0.36 | 200 | 1.3388 | 1.3359 | 0.21
135 | 1.6361 | 1.6305 | 0.34 |  |  |  |

Ref. [24] used these analytic orbitals to first construct the broken symmetry
UHF determinant, $\Psi^{\rm{UHF}}_{N}$, and then proceeded to derive analytic
expressions for the many-body REM wave functions by applying onto
$\Psi^{\rm{UHF}}_{N}$ an appropriate projection operator ${\cal P}_{L}$ (see
section 2.2.1) that restores the circular symmetry and generates correlated
wave functions with good total angular momentum $L$. These REM wave functions
can be easily written down [24] in second-quantized form for any classical
polygonal ring arrangement $(n_{1},n_{2},\ldots,n_{r})$ by following certain
simple rules for determining the coefficients of the determinants
$D(l_{1},l_{2},\ldots,l_{N})\equiv{\rm{det}}[z_{1}^{l_{1}},z_{2}^{l_{2}},\cdot\cdot\cdot,z_{N}^{l_{N}}]$,
where the $l_{j}$’s denote the angular momenta of the individual electrons.

The REM functions associated with the $(0,N)$ and $(1,N-1)$ ring arrangements,
respectively [here $(0,N)$ denotes a regular polygon with $N$ vertices, such
as an equilateral triangle or a regular hexagon, and $(1,N-1)$ is a regular
polygon with $N-1$ vertices and one occupied site in its center], are given by

$\displaystyle\Phi_{L}^{(0,N)}(z_{1},z_{2},\ldots,z_{N})$
$\displaystyle=\sum^{l_{1}+\cdot\cdot\cdot+l_{N}=L}_{0\leq
l_{1}<l_{2}<\cdot\cdot\cdot<l_{N}}\left(\prod_{i=1}^{N}l_{i}!\right)^{-1}\left(\prod_{1\leq
i<j\leq N}\sin\left[\frac{\pi}{N}(l_{i}-l_{j})\right]\right)$ (6.2)
$\displaystyle\times\;D(l_{1},l_{2},\ldots,l_{N})\exp(-\sum_{i=1}^{N}z_{i}z_{i}^{*}/2),$

with

$L=L_{0}+Nm,\;\;m=0,1,2,3,\ldots,$ (6.3)

and

$\displaystyle\Phi_{L}^{(1,N-1)}(z_{1},z_{2},\ldots,z_{N})$
$\displaystyle=\sum^{l_{2}+\cdot\cdot\cdot+l_{N}=L}_{1\leq
l_{2}<l_{3}<\cdot\cdot\cdot<l_{N}}\left(\prod_{i=2}^{N}l_{i}!\right)^{-1}\left(\prod_{2\leq
i<j\leq N}\sin\left[\frac{\pi}{N-1}(l_{i}-l_{j})\right]\right)$ (6.4)
$\displaystyle\times\;D(0,l_{2},\ldots,l_{N})\exp(-\sum_{i=1}^{N}z_{i}z_{i}^{*}/2),$

with

$L=L_{0}+(N-1)m,\;\;m=0,1,2,3,\ldots,$ (6.5)

where $L_{0}=N(N-1)/2$ is the minimum allowed total angular momentum for $N$
(fully spin polarized) electrons in high magnetic fields.

Notice that the REM wave functions [Equation (6.2) and Equation (6.4)] vanish
identically for values of the total angular momenta outside the specific
values given by the sequences (6.3) and (6.5), respectively; these sequences
are termed as magic angular momentum sequences.

Figure 17:  Total interaction energy from exact-diagonalization calculations
as a function of the total angular momentum $(19\leq L\leq 140)$ for $N=6$
electrons in the lowest Landau level. The upwards pointing arrows indicate the
magic angular momenta corresponding to the classically most stable (1,5)
polygonal ring arrangement of the Wigner molecule. The short downwards
pointing arrows indicate successful predictions of the composite-fermion
model. The medium-size downwards pointing arrows indicate predictions of the
composite-fermion model that fail to materialize as magic angular momenta. The
long downward arrows indicate EXD magic angular momenta not predicted by the
composite-fermion model. Energies in units of $e^{2}/\kappa l_{B}$, where
$\kappa$ is the dielectric constant.

We remark that, while the original REM analytic wave function was derived in
the context of a high magnetic field (that is in the fractional quantum Hall
effect regime), it is valid for any circumstance where the spectrum consists
of a degenerate manifold of LLL-like states (even with no magnetic field
present). Indeed a wave function having the form of the REM wave function
discussed by us above has been employed recently for graphene quantum dots
with a zig-zag boundary condition and in the absence of a magnetic field
[231].

In the remaining of this section, we continue discussing the properties of
analytic REM wave functions associated with fully spin polarized electrons.
However, we mention here that, following the methodology of Ref. [24] for
fully spin polarized REMs. Dai et al[232] and Shi et al[233] have most
recently presented analytic trial wave functions for rotating electron
molecules with partial spin polarizations.

###  6.2 Yrast rotational band in the lowest Landau level

As an accuracy test, we compare in table 1 REM and exact-diagonalization
results for the interaction energies of the yrast band associated with the
magic angular momenta $L_{m}$ [see (2.26)] of $N=6$ electrons in the lowest
Landau level. An yrast888The word yrast is the superlative of the Swedish yr,
which means dizzy [11]. The term yrast is widely used in nuclear spectroscopy.
state is defined as the lowest-energy state for a given angular momentum $L$.
As a result, the yrast band represents excitations with purely rotational
motion; no other excitations, like center-of-mass or vibrational modes, are
present.

As seen from table 1, the REM wave functions offer an excellent approximation
to the EXD ones, since the relative error of the REM energies is smaller than
0.3%, and it decreases steadily for larger $L$ values. Of course, a small
difference in the energies between approximate and exact-diagonalization
results is only one of several tests for deciding whether a given trial wave
function is a good approximation. As will be discussed below, comparison of
conditional probability distrubutions is an equally (if not more) important
test.

###  6.3 Inconsistencies of the composite-fermion view for semiconductor
quantum dots.

Before the development of the REM approach, electrons in the lowest Landau
level in two-dimensional quantum dots were thought as being well approximated
by composite fermion trial wave functions. However, results obtained with the
REM and exact-diagonalization calculations led researchers to examine
inconsistencies and discrepancies of the CF approach in the context of quantum
dots. This section focuses on these issues.

For $N=6$, figure 17 displays (in four frames) the total interaction energy
from exact-diagonalization as a function of the total angular momentum $L$ in
the range $19\leq L\leq 140$. [The total kinetic energy in the Hamiltonian
(4.1), being a constant, can be disregarded.] One can immediately observe the
appearance of downward cusps, implying states of enhanced stability, at
certain magic angular momenta.

For the CF theory, the magic angular momenta can be determined by

$L=L^{*}+mN(N-1)=L^{*}+2mL_{0}.$ (6.6)

Namely, for $N=6$, if one knows the non-interacting $L^{*}$’s, the CF magic
$L$’s in any filling-factor interval $1/(2m-1)\geq\nu\geq 1/(2m+1)$
[corresponding to the angular-momentum interval $15(2m-1)\leq L\leq
15(2m+1)]$, $m=1,2,3,4,\ldots$, can be found by adding $2mL_{0}=30m$ units of
angular momentum to each of the $L^{*}$’s. To obtain the non-interacting
$L^{*}$’s, one needs first to construct [26, 131, 229] the compact Slater
determinants. Let $N_{n}$ denote the number of electrons in the $n$th Landau
level with $\sum_{n=0}^{t}N_{n}=N$; $t$ is the index of the highest occupied
Landau level and all the lower Landau levels with $n\leq t$ are assumed to be
occupied. The compact determinants are defined as those in which the $N_{n}$
electrons occupy contiguously the single-particle orbitals (of each $n$th
Landau level) having the lowest angular momenta $l=-n,-n+1,\ldots,-n+N_{n}-1$.
The compact Slater determinants are usually denoted as
$[N_{0},N_{1},\ldots,N_{t}]$; see Refs. [25, 229] for details.

The compact determinants and the corresponding non-interacting $L^{*}$’s for
$n=6$ are listed in Table 2.

Table 2:  Compact Slater determinants and associated angular momenta $L^{*}$
for $N=6$ electrons according to the CF presciption. Both $L^{*}=-3$ and
$L^{*}=3$ are associated with two compact states each, the one with lowest
energy being the preferred one.

  * Compact state | $L^{*}$
---|---
$[$1,1,1,1,1,1$]$ |  $-$15
$[$2,1,1,1,1$]$ |  $-$9
$[$2,2,1,1$]$ |  $-$5
$[$3,1,1,1$]$ |  $-$3
$[$2,2,2$]$ |  $-$3
$[$3,2,1$]$ | 0
$[$4,1,1$]$ | 3
$[$3,3$]$ | 3
$[$4,2$]$ | 5
$[$5,1$]$ | 9
$[$6$]$ | 15

There are nine different values of $L^{*}$’s, and thus the CF theory for $N=6$
predicts that there are always nine magic numbers in any interval
$15(2m-1)\leq L\leq 15(2m+1)$ between two consecutive angular momenta of
Jastrow/Laughlin states, $15(2m-1)$ and $15(2m+1)$, $m=1,2,3,\ldots$
(henceforth we will denote this interval as ${\cal I}_{m}$). For example,
using Table 2 and (6.6), the CF magic numbers in the interval $15\leq L\leq
45$ ($m=1$) are found to be the following nine

$15,\;21,\;25,\;27,\;30,\;33,\;35,\;39,\;45.$ (6.7)

On the other hand, in the interval $105\leq L\leq 135$ ($m=4$), the CF theory
predicts the following set of nine magic numbers,

$105,\;111,\;115,\;117,\;120,\;123,\;125,\;129,\;135.$ (6.8)

An inspection of the total-energy-vs-$L$ plots in figure 17 reveals that the
CF prediction misses the actual magic angular momenta specified by the exact-
diagonalization calculations as those associated with the downward cusps. It
is apparent that the number of downward cusps in any interval ${\cal I}_{m}$
is always different from 9. Indeed, there are 10 cusps in ${\cal I}_{1}$
[including that at $L=15$, not shown in figure 17(a)], 10 in ${\cal I}_{2}$
[see figure 17(b)], 7 in ${\cal I}_{3}$ [see figure 17(c)], and 7 in ${\cal
I}_{4}$ [see figure 17(d)]. In detail, the CF theory fails in the following
two aspects: (I) There are exact magic numbers that are consistently missing
from the CF prediction in every interval; with the exception of the lowest
$L=20$, these exact magic numbers (marked by a long downward arrow in the
figures) are given by $L=10(3m-1)$ and $L=10(3m+1)$, $m=1,2,3,4,\ldots$; (II)
There are CF magic numbers that do not correspond to downward cusps in the EXD
calculations (marked by medium-size downward arrows in the figures). This
happens because cusps associated with $L$’s whose difference from $L_{0}$ is
divisible by 6 (but not simultaneously by 5) progressively weaken and
completely disappear in the intervals ${\cal I}_{m}$ with $m\geq 3$; only
cusps with the difference $L-L_{0}$ divisible by 5 survive. On the other hand,
the CF model predicts the appearance of four magic numbers with $L-L_{0}$
divisible solely by 6 in every interval ${\cal I}_{m}$, at $L=30m\mp 9$ and
$30m\mp 3$, $m=1,2,3,\ldots$ The overall extent of the inadequacy of the CF
model can be appreciated better by the fact that there are six false
predictions (long and medium-size downward arrows) in every interval ${\cal
I}_{m}$ with $m\geq 3$, compared to only five correct ones [small downward
arrows, see figure 17(c) and figure 17(d)].

In contrast to the CF model, the magic angular momenta in the REM theory are
associated with the polygonal ring configurations of $N$ classical point
charges. This is due to the fact that the enhanced stability of the downward
cusps results from the coherent collective rotation of the regular-polygon REM
structures. Due to symmetry requirements, such collective rotation can take
place only at magic-angular-momenta values. The in-between angular momenta
require the excitation of additional degrees of freedom (like the center of
mass and/or vibrational modes), which raises the total energy with respect to
the values associated with the magic angular momenta.

For $N=6$, the ring configuration of lowest energy is the (1,5), while there
exists a (0,6) isomer [114, 222] with higher energy. As a result, our exact-
diagonalization calculations [26] (as well as earlier ones [131, 132, 171] for
lower angular momenta $L\leq 70$) have found that there exist two sequences of
magic angular momenta, a primary one $(S_{p})$ with $L=15+5m$ [see Equation
(6.5)], associated with the most stable (1,5) classical molecular
configuration, and a secondary one $(S_{s})$ with $L=15+6m$ [see Equation
(6.3)], associated with the metastable $(0,6)$ ring arrangement. Furthermore,
our calculations (see also Refs. [132, 171]) show that the secondary sequence
$S_{s}$ contributes only in a narrow range of the lowest angular momenta; in
the region of higher angular momenta, the primary sequence $S_{p}$ is the only
one that survives and the magic numbers exhibit a period of five units of
angular momentum. It is interesting to note that the initial competition
between the primary and secondary sequences, and the subsequent prevalence of
the primary one, has been seen in other sizes as well [171] i.e., $N=5,7,8$.
Furthermore, this competition is reflected in the field-induced molecular
phase transitions associated with broken symmetry UHF solutions in a parabolic
QD. Indeed, Ref. [53] demonstrated recently that, as a function of increasing
$B$, the UHF solutions for $N=6$ first depict the transformation of the
maximum density droplet [126] (see definition in section 2.2.1) into the (0,6)
molecular configuration; then (at higher $B$) the (1,5) configuration replaces
the (0,6) structure as the one having the lower HF energy.

The extensive comparisons in this section lead to the conclusion that the
composite-fermion model does not explain the systematic trends exhibited by
the magic angular momenta in 2D quantum dots in high magnetic fields. These
trends, however, were shown to be a natural consequence of the formation of
REMs and their metastable isomers.

These results motivated a reexamination of the original composite-fermion
approach (the mean-field CF) and led to a reassessment of the significance of
the residual interaction, neglected in the mean-field CF theory. Initially, it
has been reported that some CF functions away from the main fractions [e.g.,
for $N=19$ and $L=1845$ and $N=19$ and $L=3555$] may reproduce the
aforementioned crystalline patterns [234].

Subsequently, Jain and coworkers have found that inclusion of the residual
interaction is absolutely necessary to account for the full range of
inconsistencies of the mean-field CF theory [176]. However, this latter
development was achieved with the trade off of abandoning the fundamental
nature of the composite fermion as an elementary, independent and weakly
interacting quasi-particle. Indeed, the revised [176] CF picture amounts to an
exact diagonalization method which uses a correlated basis set (made out of CF
wave functions).

Another attempt to update the CF theory in order to account for
crystallization consists of combining the REM analytic wave function
$\Phi^{\rm REM}_{L}(z_{1},z_{2},\ldots,z_{N})$ (see section 6.1) with Jastrow
prefactors [235], namely one uses a variational wave function of the form

$\Psi^{{2p,}{\rm CFC}}_{L}=\prod_{i<j}(z_{i}-z_{j})^{2p}\Phi^{\rm
REM}_{L^{*}},$ (6.9)

with $L=L^{*}+pN(N-1)$ and $p$ serves as an additional variational parameter.
Obviously, the crystalline patterns in such an approach originate from the REM
wave function and the Jastrow prefactors simply increase the variational
freedom, leading to a numerical improvement. Although this approach is a
straightforward variational improvement of the analytic REM method [24], it is
being referred to [233, 235] as a composite-fermion crystal (CFC).

More direct variational improvements of the analytic REM wave functions can be
devised in the spirit of the variation-after-projection method. For example,
one can use angular-momentum conserving variational parameters in front of the
sine coefficients in the REM expansion [231].

Figure 18: (Color online) Conditional probability distributions at high $B$
for $N=6$ electrons and $L=75$ ($\nu=1/5$, left column) and for $N=7$
electrons and $L=105$ (again $\nu=1/5$, right column). Top row: REM case.
Middle row: The case of exact diagonalization. Bottom row: The
Jastrow/Laughlin case. The exact diagonalization and REM wave functions have a
pronouned crystalline character, corresponding to the (1,5) polygonal
configuration of the REM for $N=6$, and to the (1,6) polygonal configuration
for $N=7$. In contrast, the Jastrow/Laughlin wave functions exhibit a
characteristic liquid profile that depends smoothly on the number $N$ of
electrons. The observation point (identified by a solid dot) is located at
$r_{0}=5.431l_{B}$ for $N=6$ and $L=75$ and $r_{0}=5.883l_{B}$ for $N=7$ and
$L=105$. The EXD Coulomb interaction energies (lowest Landau level) are 2.2018
and 2.9144 $e^{2}/\kappa l_{B}$ for $N=6,L=75$ and $N=7,L=105$, respectively.
The errors relative to the corresponding exact-diagonalization energies and
the overlaps of the trial functions with the EXD ones are: (I) For $N=6,L=75$,
REM: 0.85%, 0.817; JL: 0.32%, 0.837. (II) For $N=7,L=105$, REM: 0.59%, 0.842;
JL: 0.55%, 0.754.  Figure 19: (Color online) CPDs at high $B$ for $N=7$ and
$L=63$ ($\nu=1/3$). Top: REM case; Middle: EXD case; Bottom: JL case. Unlike
the JL CPD (which is liquid), the CPDs for the exact-diagonalization and REM
wave functions exhibit a well developed crystalline character [corresponding
to the (1,6) polygonal configuration of the REM for $N=7$ electrons]. The
observation point (identified by a solid dot) is located at
$r_{0}=4.568l_{B}$.

###  6.4 REM versus Laughlin wave functions: Conditional probability
distributions and multiplicity of zeroes

Recent extensive numerical calculations [24, 52] have revealed major
disagreements between the intrinsic structure of the Jastrow/Laughlin trial
wave functions [55] for the main fractions $\nu=1/(2m+1)$ and that of the
exact-diagonalization and REM wave functions. Indeed, it was found that both
EXD and REM wave functions exhibit crystalline correlations, while the
Jastrow/Laughlin ones are liquid-like as originally described in Ref. [55].

To illustrate the differences between the intrinsic structure of the REM and
EXD states in the lowest Landau level versus the familiar Jastrow/Laughlin
ones, we display in figure 18 the CPDs for cusp states corresponding to a low
filling factor $\nu=1/5$ and for two different sizes, i.e., for $N=6$
electrons ($L=75$, left column) and $N=7$ electrons ($L=105$, right column).
In figure 18, the top row depicts the REM case; the EXD case is given by the
middle row, while the CF case [which reduces to the JL wave functions for
fractions $1/(2p+1)$] are given by the bottom row.

There are three principal conclusions that can be drawn from an inspection of
figure 18 (and the many other cases studied in Ref. [26]).

(I) The character of the exact-diagonalization states is unmistakably
crystalline with the EXD CPDs exhibiting a well developed molecular polygonal
configuration [(1,5) for $N=6$ and (1,6) for $N=7$, with one electron at the
center], in agreement with the explicitly crystalline REM case.

(II) For all the examined instances covering the low fractional fillings 1/9,
1/7, and 1/5, the Jastrow/Laughlin wave functions fail to capture the
intrinsic crystallinity of the exact-diagonalization states. In contrast, they
represent “liquid” states in agreement with an analysis that goes back to the
original papers [55, 236] by Laughlin. In particular, Ref. [236] investigated
the character of the JL states through the use of a pair correlation function
[usually denoted by $g(R)$] that determines the probability of finding another
electron at the absolute relative distance $R=|{\bf r}-{\bf r}_{0}|$ from the
observation point ${\bf r}_{0}$. Our anisotropic CPD of Equation (1.1) is of
course more general (and more difficult to calculate) than the $g(R)$ function
of Ref. [236]. However, both our $P({\bf r},{\bf r}_{0})$ (for $N=6$ and $N=7$
electrons) and the $g(R)$ (for $N=1000$ electrons, and for $\nu=1/3$ and
$\nu=1/5$) in Ref. [236] reveal a similar characteristic liquid-like and
short-range-order behavior for the JL states, eloquently described in Ref.
[236] (see p. 249 and p. 251). Indeed, we remark that only the first-neighbor
electrons on the outer rings can be distinguished as separate localized
electrons in our CPD plots of the JL functions [see figure 18].

(III) For a finite number of electrons, pronounced crystallinity of the exact-
diagonalization states occurs already at the $\nu=1/5$ value (see figure 18).
This finding is particularly interesting in light of expectations [234, 237]
(based on comparisons [55, 236, 238] between the JL states and the static bulk
Wigner crystal) that a liquid-to-crystal phase transition may take place only
at lower fillings with $\nu\leq 1/7$.

Of interest also is the case of $\nu=1/3$. Indeed, for this fractional
filling, the liquid JL function is expected to provide the best approximation,
due to very high overlaps (better than 0.99) with the exact wave function [58,
59, 161]. In figure 19, we display the CPDs for $N=7$ and $L=63$ ($\nu=1/3$),
and for the three cases of REM, EXD, and JL wave functions. Again, even in
this most favorable case, the CPD of the JL function disagrees with the EXD
one, which exhibits clearly a (1,6) crystalline configuration in agreement
with the REM CPD.

Similar crystalline correlations at higher fractions were also found for
quantum dots of larger sizes, e.g., $N=8$, and $N=9$ electrons. As
illustrative examples for these additional sizes [see also the EXD CPD for
$N=12$ electrons in figure 24 below (in section 7.3)], we displayed in figure
5 of Ref. [26] the CPDs for $N=8$ and $L=91$ ($1/5<\nu=4/13<1/3$) and for
$N=9$ and $L=101$ ( $1/3<\nu=36/101<1$). Again, the CPDs (both for the REM and
the EXD wave functions) exhibit a well developed crystalline character in
accordance with the (1,7) and (2,7) polygonal configurations of the REM,
appropriate for $N=8$ and $N=9$ electrons, respectively.

Another area of disagreement between REM and Laughlin wave functions concerns
the properties of the zero points. In this respect, we recall that the
Jastrow/Laughlin trial functions for $N$ electrons have the form

$\Phi^{\rm JL}(z_{1},z_{2},\ldots,z_{N})=\left(\prod_{1\leq i<j\leq
N}(z_{i}-z_{j})^{2m+1}\right)\exp(-\sum_{i=1}^{N}z_{i}z_{i}^{*}/2).$ (6.10)

Due to the Jastrow factors $(z_{i}-z_{j})^{2m+1}$, it is apparent that the
Laughlin expressions (6.10) (as a function of a given $z_{i}$) have $N-1$ zero
points, each of order $2m+1$, which are bound to the positions of the
remaining $N-1$ electrons. In contrast, as discussed in Ref. [24], the
analytic REM wave functions do not have zeroes with order higher than unity.
In particular, only $N-1$ of the REM zeroes are bound to the positions of the
remaining electrons, while the rest of them are free. Recently, it has been
shown through extensive numerical studies [239] that the properties of REM
zeroes are in agreement with the behavior of the zeroes in exact-
diagonalization wave functions; this is another indication of the superiority
of the REM picture compared to the Laughlin theory.

Before exiting this discussion, we remark about discrepancies of the Laughlin
quasihole theory in the context of quantum dots. In particular, we recall that
the Laughlin quasihole, with $N$ additional units of angular momentum, has
been conjectured to be the first excited state. However, LLL exact-
diagonalization calculations for $N$ electrons in a quantum dot have revealed
that this is not the case. Instead, the first excited state corresponds to an
increment in the total angular momentum which varies as the number of
electrons localized on one of the rings of the rotating electron molecule,
usually the outermost one; see figure 26 in section 7.4 below.

##  7 Rotating electron molecules in two-dimensional quantum dots under a
strong, but finite external magnetic field ($\omega_{c}/2\omega_{0}>1$)

Figure 20: (Color online) Two-step-method versus exact-diagonalization
calculations: Ground-state energies for $N=4$ electrons (referenced to
$4\hbar\tilde{\omega}$) as a function of the magnetic field $B$. Thick dashed
line (red): broken-symmetry UHF (static electron molecule). Solid line
(green): EXD (from Ref. [171]). Thick dashed-dotted line (blue): REM. Thin
dashed line (violet, marked LLL): the commonly used approximate energies
$\tilde{E}^{\rm{EXD}}_{\rm{LLL}}(B)$ (see text for details). Thin dotted line
(black): $\tilde{E}^{\rm{REM}}_{\rm{LLL}}(B)$ (see text). For $B<8$ T, the
$\tilde{E}^{\rm{EXD}}_{\rm{LLL}}(B)$ and $\tilde{E}^{\rm{REM}}_{\rm{LLL}}(B)$
curves coincide; we have checked that these curves approach each other also at
larger values of $B$, outside the plotted range. Numbers near the bottom
curves denote the value of magic angular momenta [$L_{m}$, see (2.26)] of the
ground state. Corresponding fractional filling factors are specified by
$\nu=N(N-1)/(2L_{m})$. Parameters used: confinement $\hbar\omega_{0}=3.60$
meV, dielectric constant $\kappa=13.1$, effective mass $m^{*}=0.067m_{e}$.

###  7.1 Ground-state energies in medium and high magnetic field

The general form (2.24) for the displaced Gaussian orbitals [in conjunction
with the projected REM wave function (2.25)] enables us to calculate REM
ground-state energies for moderately-high $B$, when corrections arising from
higher Landau levels must be taken into consideration. Unlike the lowest-
Landau-level case, where the azimuthal integration can be carried out
analytically, the energies (2.28) (and corresponding CPDs) associated with the
general REM wave function (2.25) require numerical integration over the
azimuthal angles $\gamma_{q}$.

Before proceeding with the presentation of results for $N>10$, we demonstrate
the accuracy of the two-step method embodied in Equation (2.25) through
comparisons with existing exact-diagonalization results for smaller sizes. In
figure 20, our REM calculations for the ground-state energies as a function of
$B$ are compared to EXD calculations [171] for $N=4$ electrons in an external
parabolic confinement. The thick dotted line (red) represents the broken-
symmetry UHF approximation (first step of our method), which naturally is a
smooth curve lying above the EXD one [solid line (green)]. The results
obtained after restoration of symmetry [dashed-dotted line (blue); marked as
REM] agree very well with the EXD one in the whole range 2 T$<B<$ 15 T. We
recall here that, for the parameters of the quantum dot, the electrons form in
the intrinsic frame of reference a square about the origin of the dot, i.e., a
(0,4) configuration, with the zero indicating that no electron is located at
the center. According to (2.26), $L_{0}=6$, and the magic angular momenta are
given by $L_{m}=6+4k$, $k=0$, 1, 2, $\ldots$ Note that the REM energies are
slightly lower than the EXD ones in several subranges. According to the
Rayleigh-Ritz variational theorem, this indicates that the hyperspherical-
harmonics calculation (equivalent to an exact-diagonalization approach) of
[171] did not converge fully in these subranges.

To further evaluate the accuracy of the two-step method, we also display in
figure 20 [thin dashed line (violet)] ground-state energies
$\tilde{E}^{\rm{EXD}}_{\rm LLL}(B)$ calculated with the commonly used
approximate LLL Hamiltonian [128, 229, 237, 240]

$\tilde{\cal H}_{\rm
LLL}=N\hbar\tilde{\omega}+\hbar(\tilde{\omega}-\frac{\omega_{c}}{2})L+\sum_{j>i=1}^{N}\frac{e^{2}}{\kappa
r_{ij}},$ (7.1)

where $\tilde{\omega}=\sqrt{\omega_{0}^{2}+\omega_{c}^{2}/4}$. The LLL
Hamiltonian $\tilde{\cal H}_{\rm LLL}$ reduces to the previously introduced
Hamiltonian ${\cal H}_{\rm LLL}$ [see Equation (4.1)] in the limit
$B\rightarrow\infty$. Both Hamiltonians restrict the many-body wave functions
within the lowest Landau level, and they both accept the same set of
eigenstates as solutions. Indeed the term
$\hbar(\tilde{\omega}-\omega_{c}/2)L$ is proportional to the total angular
momentum, and thus its presence influences only the eigenvalues, but not the
composition of the eigenstates. $\tilde{\cal H}_{\rm LLL}$ corresponds to a
situation where the external harmonic confinement is added to ${\cal H}_{\rm
LLL}$ as a perturbation (see section II.B in Ref. [53]). As a result, (i) the
degeneracy of the single-particle levels in the lowest Landau level is lifted
and (ii) there is an eigenstate with minimum energy (the ground state) at each
value of $B$ (expressed through the cyclotron frequency $\omega_{c}$).
Naturally, the LLL levels used in the exact diagonalization of $\tilde{\cal
H}_{\rm LLL}$ are given by expression (4.2), but with
$\Lambda=\tilde{l}=\sqrt{\hbar/(m^{*}\tilde{\omega})}$.

We find that the energies $\tilde{E}^{\rm EXD}_{\rm LLL}(B)$ tend to
substantially overestimate the REM (and EXD) energies for lower values of $B$
(e.g., by as much as 5.5% at $B\sim 4$ T). On the other hand, for higher
values of $B$ ($>$ 12 T), the energies $\tilde{E}^{\rm{EXD}}_{\rm{LLL}}(B)$
tend to agree rather well with the REM ones. We stress that the results
labelled simply as EXD correspond to exact diagonalizations without any
restrictions on the Hilbert space, i.e., the full Darwin-Fock single-particle
spectrum is considered at a given $B$.

A behavior similar to $\tilde{E}^{\rm EXD}_{\rm LLL}(B)$ is exhibited also by
the $\tilde{E}^{\rm{REM}}_{\rm{LLL}}(B)$ ground-state energies [which are
calculated using the Hamiltonian (7.1) and the LLL analytic REM wave functions
in section 6.1 with lengths in units of $\sqrt{\hbar/(m^{*}\tilde{\omega})}$
instead of $l_{B}\sqrt{2}$; dotted line (black)]. A similar agreement between
REM and EXD results, and a similar inaccurate behavior of the LLL approximate
Hamiltonian (7.1) was found by us also for $N=3$ electrons in the range 2 T
$<B<$ 16 T shown in figure 2 of Ref. [53] (the exact-diagonalization
calculation in this figure was taken from Ref. [166]).

In all cases, the total energy of the REM is lower than that of the UHF Slater
determinant (see, e.g., figure 20). Indeed, a theorem discussed in section 3
of Ref. [241], pertaining to the energies of projected wave functions,
guarantees that this lowering of energy applies for all values of $N$ and $B$.

###  7.2 The case of $N=11$ electrons.

Figure 21: (Color online) Ground-state energies for $N=11$ electrons (per
particle, referenced to $\hbar\tilde{\omega}$) as a function of the magnetic
field $B$. Dashed line (red): UHF (static electron molecule). Solid line
(blue): REM. Dotted line (black): Approximate energies
$\tilde{E}^{\rm{REM}}_{\rm{LLL}}(B)$ (see text). Parameters used: confinement
$\hbar\omega_{0}=3.60$ meV, dielectric constant $\kappa=13.1$, effective mass
$m^{*}=0.067m_{e}$. The inset shows a magnification of the REM curve in the
range 5 T $<B<$ 12 T.  Table 3: Ground-state magic angular momenta and their
decomposition $\\{k_{1},k_{2}\\}$ for $N=11$ in the nagnetic-field range 5 T
$\leq B\leq$ 25 T. The results correspond to the REM (see lower curve in
figure 21). The parameters used are as in figure 21.

  * $L_{m}$ | $k_{1}$ | $k_{2}$ | $L_{m}$ | $k_{1}$ | $k_{2}$
---|---|---|---|---|---
55 | 0 | 0 | 165 | 2 | 13
63 | 0 | 1 | 173 | 2 | 14
71 | 0 | 2 | 181 | 2 | 15
79 | 0 | 3 | 189 | 2 | 16
90 | 1 | 4 | 197 | 2 | 17
98 | 1 | 5 | 205 | 2 | 18
106 | 1 | 6 | 213 | 2 | 19
114 | 1 | 7 | 224 | 3 | 20
122 | 1 | 8 | 232 | 3 | 21
130 | 1 | 9 | 240 | 3 | 22
138 | 1 | 10 | 248 | 3 | 23
146 | 1 | 11 | 256 | 3 | 24
154 | 1 | 12 |  |  |

Figure 21 presents the case for the ground-state energies of a quantum dot
with $N=11$ electrons, which have a nontrivial double-ring configuration
$(n_{1},n_{2})$. The most stable [114] classical configuration is $(3,8)$, for
which we have carried UHF (static electron molecule) and REM (projected)
calculations in the magnetic field range 5 T $<B<$ 25 T. Figure 21 also
displays the LLL ground-state energies $\tilde{E}^{\rm{REM}}_{\rm{LLL}}(B)$
[dotted curve (black)], which, as in previous cases, overestimate the ground-
state energies for smaller $B$. The approximation
$\tilde{E}^{\rm{REM}}_{\rm{LLL}}(B)$, however, can be used to calculate
ground-state energies for higher values of $B$. In keeping with the findings
for smaller sizes [51] [with $(0,N)$ or $(1,N-1)$ configurations], we found
that both the UHF and the REM ground-state energies approach, as
$B\rightarrow\infty$, the classical equilibrium energy of the (3,8) polygonal
configuration [i.e., 19.94 meV; 4.865$E_{0}$ in the units of Ref. [114],
$E_{0}\equiv(m^{*}\omega_{0}^{2}e^{4}/2\kappa^{2})^{1/3}$].

In analogy with smaller sizes (see, e.g., figure 20 and Ref. [53]), the REM
ground-state energies in figure 21 exhibit oscillations as a function of $B$
(see in particular the inset). These oscillations are associated with magic
angular momenta, specified by the number of electrons on each ring. For $N=11$
they are given by (2.26), i.e., $L_{m}=55+3k_{1}+8k_{2}$, with the $k_{q}$’s
being nonnegative integers. As was the case with $N=9$ electrons [53], an
analysis of the actual values taken by the set of indices $\\{k_{1},k_{2}\\}$
reveals several additional trends that further limit the allowed values of
ground-state $L_{m}$’s. In particular, starting with the values $\\{0,0\\}$ at
$B=5$ T ($L_{0}=55$), the indices $\\{k_{1},k_{2}\\}$ reach the values
$\\{3,24\\}$ at $B=25$ T ($L_{m}=256$). As seen from table 3, the outer index
$k_{2}$ changes faster than the inner index $k_{1}$. This behavior minimizes
the total kinetic energy of the independently rotating rings; indeed, the
kinetic energy of the inner ring (as a function of $k_{1}$) rises faster than
that of the outer ring (as a function of $k_{2}$) due to smaller moment of
inertia (smaller radius) of the inner ring [see (7.2)].

Figure 22: (Color online) Conditional probability distributions for the REM
ground state of $N=11$ electrons at $B=10$ T ($L=106$). The electrons are
arranged in a (3,8) structure. The observation point (solid dot) is placed on
(left) the outer ring at $r_{0}=1.480R_{0}$, and (right) on the inner ring at
$r_{0}=0.557R_{0}$. Parameters used: confinement $\hbar\omega_{0}=3.60$ meV,
dielectric constant $\kappa=13.1$, effective mass $m^{*}=0.067m_{e}$. Lengths
in units of $R_{0}=(2e^{2}/m^{*}\kappa\omega_{0}^{2})^{1/3}$. CPDs (vertical
axes) in arbitrary units.

In addition to the overestimation of the ground-state energy values for
smaller magnetic fields (see figure 21 and our discussion above), there are
additional shortcomings of the lowest-Landau-level approximation pertaining to
the ground-state ring configurations. In particular, for $N=11$, we find that
according to the LLL approximation the ground-state angular momentum
immediately after the maximum density droplet ($L_{0}=55$) is $L_{m}=66$,
i.e., the one associated with the $(0,N)$ vortex-in-the-center configuration.
This result, erroneously stated in Refs. [242, 243] as the ground state,
disagrees with the correct result that includes the full effect of the
confinement and is listed in table 3, where the ground-state angular momentum
immediately following the maximum density droplet is $L_{m}=63$. This angular-
momentum value corresponds to the classicaly most stable (3,8) ring
configuration, that is, a configuration with no vortex at all (see also the
case of $N=9$ electrons in Ref. [53]).

Figure 22 displays the REM conditional probability distributions for the
ground state of $N=11$ electrons at $B=10$ T ($L_{m}=106$). The (3,8) ring
configuration is clearly visible. We note that when the observation point is
placed on the outer ring (left panel), the CPD reveals the crystalline
structure of this ring only; the inner ring appears to have a uniform density.
To reveal the crystalline structure of the inner ring, the observation point
must be placed on this ring; then the outer ring appears to be uniform in
density. This behavior suggests that the two rings rotate independently of
each other, a property that is explored in the next section to derive an
approximate quasiclassical expression for the yrast rotational spectra
associated with an arbitrary number of electrons.

###  7.3 Approximate analytic expression for the yrast-band spectra

In figure 23, we display the CPD for the REM wave function of $N=17$
electrons. This case has a nontrivial three-ring structure (1,6,10) [114]
which is sufficiently complex to allow generalizations for larger numbers of
particles. The remarkable floppy character (leading to a non-classical, non-
rigid rotational inertia, see section VI of Ref. [53]) of the REM is
illustrated in the CPDs of figure 23. Indeed, as the two CPDs [reflecting the
choice of taking the observation point [${\bf r}_{0}$ in (1.1)] on the outer
(left frame) or the inner ring (right frame)] reveal, the polygonal electron
rings rotate independently of each other. Thus, e.g., to an observer located
on the inner ring, the outer ring will appear as having a uniform density, and
vice versa. The wave functions obtained from exact diagonalization exhibit
also the property of independently rotating rings [see, e.g., the $N=12$ and
$L=132$ ($\nu=1/2$) case in figure 24], which is a testimony to the ability of
the REM wave function to capture the essential physics of a finite number of
electrons in high $B$. In particular, the conditional probability distribution
displayed in figure 24 for exact-diagonalization wave functions exhibits the
characteristics expected from the CPD evaluated using REM wave functions for
the (3,9) configuration and with an angular-momentum decomposition into shell
contributions [see Equation (2.25) and Equation (2.27)] $L_{1}=3+3k_{1}$ and
$L_{2}=63+9k_{2}$ ($L_{1}+L_{2}=L_{m}$; for $L_{m}=132$ the angular-momentum
decomposition is $L_{1}=6$ and $L_{2}=126$).

Figure 23: (Color online) Ground-state conditional probability distributions
obtained from REM wave functions for the ground state of $N=17$ electrons at
$B=10$ T ($L=228$). The electrons are arranged in a (1,6,10) structure. The
observation point (solid dot) is placed on the outer ring at
$r_{0}=1.858R_{0}$ (left frame), and on the inner ring at $r_{0}=0.969R_{0}$
(right frame). The rest of the parameters are: confinement
$\hbar\omega_{0}=3.6$ meV, dielectric constant $\kappa=13.1$, effective mass
$m^{*}=0.067m_{e}$. Lengths in units of $R_{0}=(2e^{2}/(\kappa
m^{*}\omega_{0}^{2}))^{1/3}$. CPDs (vertical axes) in arbitrary units.  Figure
24: (Color online) CPDs for $N=12$ electrons and with angular momentum $L=132$
($\nu=1/2$) calculated using exact diagonalization in the lowest Landau level.
The electrons are arranged in a (3,9) structure. The observation point (solid
dot) is placed on the outer ring at $r_{0}=5.22l_{B}$ (left frame), and on the
inner ring at $r_{0}=1.87l_{B}$ (right frame). Lengths in units of $l_{B}$.
CPDs (vertical axes) in arbitrary units.

In addition to the conditional probabilities, the floppy-rotor character of
the REM is revealed in its excited rotational spectrum for a given $B$. From
our microscopic calculations based on the wave function in (2.25), we have
derived (see below) an approximate (denoted as “app”), but analytic and
parameter-free, expression [see (7.7) below] which reflects directly the
nonrigid character of the REM for arbitrary size. This expression allows
calculation of the energies of REMs for arbitrary $N$, given the corresponding
equilibrium configuration of confined classical point charges.

We focus on the description of the yrast band at a given $B$. Motivated by the
aforementioned nonrigid character of the rotating electron molecule, we
consider the following kinetic-energy term corresponding to a
$(n_{1},\ldots,n_{q},\ldots,n_{r})$ configuration (with
$\sum_{q=1}^{r}n_{q}=N$):

$E_{\rm{app}}^{\rm{kin}}(N)=\sum_{q=1}^{r}\hbar^{2}L_{q}^{2}/(2{\cal
J}_{q}(a_{q}))-\hbar\omega_{c}L/2,$ (7.2)

where $L_{q}$ is the partial angular momentum associated with the $q$th ring
about the center of the dot and the total angular momentum is
$L=\sum_{q=1}^{r}L_{q}$. ${\cal J}_{q}(a_{q}))\equiv n_{q}m^{*}a_{q}^{2}$ is
the rotational moment of inertia of each individual ring, i.e., the moment of
inertia of $n_{q}$ classical point charges on the $q$th polygonal ring of
radius $a_{q}$. To obtain the total energy, $E_{L}^{\rm{REM}}$, we include
also the term $E_{\rm{app}}^{\rm{hc}}(N)=\sum_{q=1}^{r}{\cal
J}_{q}(a_{q})\tilde{\omega}^{2}/2$ due to the effective harmonic confinement
$\tilde{\omega}$ (see Appendix A.1), as well as the interaction energy
$E_{\rm{app}}^{C}$,

$E_{\rm{app}}^{C}(N)=\sum_{q=1}^{r}\frac{n_{q}S_{q}}{4}\frac{e^{2}}{\kappa
a_{q}}+\sum_{q=1}^{r-1}\sum_{s>q}^{r}V_{C}(a_{q},a_{s}).$ (7.3)

The first term is the intra-ring Coulomb-repulsion energy of $n_{q}$ point-
like electrons on a given ring, with a structure factor

$S_{q}=\sum_{j=2}^{n_{q}}(\sin[(j-1)\pi/n_{q}])^{-1}.$ (7.4)

The second term is the inter-ring Coulomb-repulsion energy between rings of
uniform charge distribution corresponding to the specified numbers of
electrons on the polygonal rings. The expression fo $V_{C}$ is

$V_{C}(a_{q},a_{s})=n_{q}{n_{s}}e^{2}[\kappa(a_{q}^{2}+a_{s}^{2})^{1/2}]^{-1}\;{{}_{2}F_{1}}[3/4,1/4;1;4a_{q}^{2}a_{s}^{2}(a_{q}^{2}+a_{s}^{2})^{-2}],$
(7.5)

where ${{}_{2}F_{1}}$ is the hypergeometric function.

For large $L$ (and/or $B$), the radii of the rings of the rotating molecule
can be found by neglecting the interaction term in the total approximate
energy, thus minimizing only
$E_{\rm{app}}^{\rm{kin}}(N)+E_{\rm{app}}^{\rm{hc}}(N)$. One finds

$a_{q}=\lambda\sqrt{L_{q}/n_{q}},$ (7.6)

with $\lambda=\tilde{l}=\sqrt{\hbar/m^{*}\tilde{\omega}}$; i.e., the ring
radii depend on the partial angular momentum $L_{q}$, reflecting the lack of
radial rigidity. Substitution into the above expressions for
$E_{\rm{app}}^{\rm{kin}}$, $E_{\rm{app}}^{\rm{hc}}$, and $E_{\rm{app}}^{C}$
yields for the total approximate energy the final expression:

$E_{\rm{app},L}^{\rm{REM}}(N)=\hbar(\tilde{\omega}-\omega_{c}/2)L+\sum_{q=1}^{r}\frac{C_{V,q}}{L_{q}^{1/2}}+\sum_{q=1}^{r-1}\sum_{s>q}^{r}V_{C}(\lambda\sqrt{\frac{L_{q}}{n_{q}}},\lambda\sqrt{\frac{L_{s}}{n_{s}}}),$
(7.7)

where the constants

$C_{V,q}=0.25n_{q}^{3/2}S_{q}e^{2}/(\kappa\lambda).$ (7.8)

For simpler $(0,N)$ and $(1,N-1)$ ring configurations, Equation (7.7) reduces
to the expressions reported earlier [51, 128].

The floppy-rotor character of the REM under strong magnetic field is reflected
in the absence in (7.7) of a kinetic-energy term proportional to $L^{2}$. This
contrasts with the rigid-rotor behavior of an electron molecule at zero
magnetic field (see section 5.2 and Ref. [51]).

###  7.4 Possible implications for the thermodynamic limit

While our focus in this section is on the behavior of trial and exact wave
functions in (finite) quantum dots in high magnetic fields, it is natural to
inquire about possible implications of our findings to fractional-quantum-
Hall-effect systems in the thermodynamic limit.

We recall that appropriate trial wave functions for clean FQHE systems possess
a good angular momentum $L\geq L_{0}$, a property shared by both the CF/JL and
REM functions [24, 55, 57, 236]. We also recall the previous finding [55, 236]
that for large fractional fillings $\nu>1/7$, the liquid-like (and circularly
uniform) Jastrow/Laughlin function is in the thermodynamic limit energetically
favored compared to the broken-symmetry static Wigner crystal (which has no
good angular momentum); for $\nu<1/7$, the static Wigner crystal becomes lower
in energy. This finding was enabled by the simple form of the JL functions,
which facilitated computations of total energies as a function of size for
sufficiently large $N$ (e.g., $N=1000$).

A main finding of the recent literature on quantum dots is that the exact-
numerical-diagonalization wave functions of small systems ($N\leq 12$) are
crystalline in character for both low and high fractional fillings. This
finding contradicts earlier suggestions [55, 229, 236] that, for high $\nu$’s,
small systems are accurately described by the liquid-like JL wave functions
and their descendants, e.g., the composite-fermion ones. Of course, for the
same high $\nu$’s, our small-size results cannot exclude the possibility that
the CPDs of the exact solution may exhibit with increasing $N$ a transition
from crystalline to liquid character, in agreement with the JL function.
However, as of now the existence of such a transition remains an open
theoretical subject.

For the low fractions, the rotating-electron-molecule theory raises still
another line of inquiry. Due to the specific form of the REM wave functions,
computational limitations (in the so-called disk geometry that is natural to
quantum dots) prevent us at present from making extrapolations of total
energies at a given $\nu$ as $N\rightarrow\infty$. Nevertheless, from the
general theory of projection operators, one can conclude that the REM energies
exhibit a different trend compared to the JL ones, whose energies were found
[55, 236] to be higher than the static Wigner crystal. Indeed the rotating-
electron-molecule wave functions remain lower in energy than the corresponding
static crystalline state for all values of $N$ and $\nu$, even in the
thermodynamic limit. This is due to an “energy gain” theorem (see Section 3 in
Ref. [241]) stating that at least one of the projected states (i.e., the
ground state) has an energy lower than that of the original broken-symmetry
trial function (e.g., the UHF determinant), and this theorem applies for any
number of electrons $N$ and for all values of the magnetic field $B$.
Naturally, the REM wave functions will be physically relevant compared to
those of the broken-symmetry crystal at the thermodynamic limit if the energy
gain does not vanish when $N\rightarrow\infty$; otherwise, one needs to
consider the posssibility that the static crystal is the relevant physical
picture.

Figure 25: Stabilization energies $\Delta
E^{\rm{gain}}_{\rm{gs}}=E^{\rm{gs}}_{\rm{REM}}-E_{\rm{UHF}}$ for $N=6$ (dashed
curve) and $N=7$ (solid curve) fully polarized electrons in a parabolic QD as
a function of $B$. The troughs associated with the major fractional fillings
(1/3, 1/5, and 1/7) and the corresponding ground-state angular momenta
[$L=N(N-1)/(2\nu)$] are indicated with arrows. We have extended the
calculations up to $B=120$ T (not shown), and verified that $\Delta
E^{\rm{gain}}_{\rm{gs}}$ remains negative while its absolute value vanishes as
$B\rightarrow\infty$. The choice of parameters is: $\hbar\omega_{0}=3$ meV
(parabolic confinment), $m^{*}=0.067m_{e}$ (electron effective mass), and
$\kappa=12.9$ (dielectric constant).

The discussion in the above paragraph may be recapitulated by the following
question: which state is the relevant one in the thermodynamic limit
$(N\rightarrow\infty)$ – the broken-symmetry one (i.e., the static crystal) or
the symmetry restored (i.e., rotating crystal) state? This question, in the
context of bulk broken-symmetry systems, has been addressed in the early work
of Anderson [10] who concluded that the broken-symmetry state (here the UHF
static crystalline solution) can be safely taken as the effective ground
state. In arriving at this conclusion Anderson invoked the concept of
(generalized) rigidity. As a concrete example, one would expect a crystal to
behave like a macroscopic body, whose Hamiltonian is that of a heavy rigid
rotor with a low-energy excitation spectrum $L^{2}/2{\cal J}$, the moment of
inertia ${\cal J}$ being of order $N$ (macroscopically large when
$N\rightarrow\infty$). The low-energy excitation spectrum of this heavy rigid
rotor above the ground-state ($L=0$) is essentially gapless (i.e.,
continuous). Thus although the formal ground state posseses continuous
rotational symmetry (i.e., $L=0$), “there is a manifold of other states,
degenerate in the $N\rightarrow\infty$ limit, which can be recombined to give
a very stable wave packet with essentially the nature” [10] of the broken-
symmetry state (i.e., the static Wigner crystal in our case). As a consequence
of the “macroscopic heaviness” as $N\rightarrow\infty$, one has: (I) The
energy gain due to symmetry restoration (i.e., the stabilization energy
$\Delta E^{\rm{gain}}_{\rm{gs}}=E^{\rm{gs}}_{\rm{REM}}-E_{\rm{UHF}}$, see
figure 25) vanishes as $N\rightarrow\infty$, and (II) The relaxation of the
system from the wave packet state (i.e., the static Wigner crystal) to the
symmetrized one (i.e., the rotating crystal) becomes exceedingly long. This
picture underlies Anderson’s aforementioned conclusion that in the
thermodynamic limit the broken-symmetry state may be used as the effective
ground state.

Consequently, in the rest of this section we will focus on issues pertaining
to the “rigidity” of the rotating electron molecule in high magnetic fields.
In particular, using our projection method and exact diagonalization, we have
demonstrated explicitly [50, 51] that the rigid-rotor picture applies to an
$N$-electron QD only when $B=0$. In contrast, in the presence of a high
magnetic field, we found [51, 52, 53] that the electrons in the quantum dot do
not exhibit global rigidity and therefore cannot be modeled as a macroscopic
rotating crystal. Instead, a more appropriate model is that of a highly non-
rigid rotor whose moment of inertia depends strongly on the value of the
angular momentum $L$. This behavior originates from the dominance of the
magnetic field over the Coulomb repulsion.

The non-rigid rotor at high $B$ has several unique properties: (I) The ground
state has angular momentum $L_{\rm{gs}}>0$; (II) While the rotating electron
molecule does not exhibit global rigidity, it possesses azimuthal rigidity
(i.e., all electrons on a given ring rotate coherently), with the rings,
however, rotating independently of each other. Furthermore, the radii of the
rings vary for different values of $L$, unlike the case of a rigid rotor;
(III) The excitation spectra do not vary as $L^{2}$; instead they consist of
terms that vary as $aL+\sum_{q=1}^{r}b_{q}/\sqrt{L_{q}}$ (with
$\sum_{q=1}^{r}L_{q}=L$; for the precise values of the constants $a$ and $b$
see section 7.3 and Refs. [51, 53]); (IV) The angular momentum values are
given by the magic values [see section 6.1]
$L=L_{0}+\sum_{q=1}^{r}k_{q}n_{q}$, where $(n_{1},n_{2},\ldots,n_{r})$ is the
polygonal ring arrangement of the static Wigner molecule (with $n_{q}$ the
number of electrons on the $q$th ring) and $k_{1}<k_{2}<\ldots<k_{q}$ are
nonnegative integers. These magic $L$’s are associated with the cusp states
which exhibit a relative energy gain with respect to neighboring excitations.
Thus the low-energy excitation spectrum of the non-rigid rotor is not dense
and exhibits gaps due to the occurrence of the magic (cusp) states (see figure
26). Furthermore, these gaps are reflected in the oscillatory behavior of
$\Delta E^{\rm{gain}}_{\rm{gs}}$ (see, e.g., figure 25) as a function of $B$
(or $\nu$).

Figure 26:  Low-energy part of the spectrum of the parabolic QD whose
parameters are the same as those in figure 25, calculated as a function of the
angular momentum $L$ through exact diagonalization for $N=7$ electrons at a
magnetic field $B=18.8$ T. We show here the spectrum in the interval $95\leq
L\leq 115$ (in the neighborhood of $\nu=1/5$). The magic angular momentum
values corresponding to cusp states are marked (99, 105, and 111), and they
are seen to be separated from the rest of the spectrum. For the given value of
$B$, the global energy minimum (ground state) occurs for $L_{\rm{gs}}=105$,
and the gap $\Delta$ to the first excited state ($L=99$) is indicated. The
lowest energies for the different $L$’s (the yrast band) in the plotted range
are connected by a dashed line, as a guide to the eye. The zero of energy
corresponds to $7\hbar\tilde{\omega}$, where
$\tilde{\omega}=(\omega_{0}^{2}+\omega_{c}^{2}/4)^{1/2}$ and
$\omega_{c}=eB/(m^{*}c)$. The horizontal arrow denotes the energy of the
Laughlin quasihole at L=112. It is seen that the Laughlin quasihole is not the
lowest excited state, as presumed in Ref. [55].

As $N$ increases, more polygonal rings are successively added, and since the
polygonal rings rotate independently of each other (see, e.g., the case of
$N=12$ in figure 24), we expect that the non-rigid-rotor picture remains valid
even as $N\rightarrow\infty$. As a result, it is plausible to conjecture the
following properties at high $B$ in the thermodynamic limit: (I) the
oscillatory character of $\Delta E^{\rm{gain}}_{\rm{gs}}$ will maintain,
yielding nonvanishing stabilization energies at the fractional fillings $\nu$,
and (II) the low-energy excitation spectra of the system will still exhibit
gaps in the neighborhood of the magic angular momenta (see figure 26). Of
course, these conjectures need to be further supported through numerical
calculations for large $N$. Nevertheless, the above discussion indicates that
the question of which state is physically relevant for low fractions in the
thermodynamic limit at high $B$ – i.e., the broken-symmetry static crystal or
the symmetrized rotating crystal – remains open, and cannot be answered solely
following the path of Anderson as described in Ref. [10].

The rotating Wigner crystal has properties characteristic of FQHE states,
i.e., it is incompressible (connected to the presence of an excitation gap)
and carries a current (while the broken-symmetry static crystal is
insulating). Thus, we may conjecture that a transition at lower fractional
fillings from a conducting state with good circular symmetry to an insulating
Wigner crystal cannot occur spontaneously for clean systems. Therefore, it
should be possible to observe FQHE-type behavior at low fractional fillings in
a clean system – a prediction that could explain the observations of Ref.
[244], where FQHE behavior has been observed for low fractional fillings
typically associated with the formation of a static Wigner crystal. In
practice, however, impurities and defects may influence the properties of the
rotating crystal (and its excitations), depending on the magnitude of the
excitation gap (see, e.g., figure 26). Thus one of the main challenges for
observation of the fractional quantum Hall effect at such low fillings relates
to fabrication of high mobility (nearly impurity-free) samples [245]. We
remark, however, that the stabilization energy and the gap $\Delta$ (see,
e.g., figure 26) diminish as the magnetic field increases, and as a result the
impurities become more efficient in influencing the rotating Wigner crystal
for the lower fractional fillings (i.e., higher angular momenta).

##  8 Bosonic molecules in rotating traps: Original results and applications

###  8.1 Variational description of rotating boson molecules

Recent experimental advances in the field of trapped ultracold neutral bosonic
gases have enabled control of the strength of interatomic interactions over
wide ranges [85, 86, 87, 246], from the very weak to the very strong. This
control is essential for experimental realizations of novel states of matter
beyond the well known Bose-Einstein condensate [85, 86, 87]. In this context,
the linear 1D Tonks-Girardeau regime of impenetrable trapped bosons has
generated intensive theoretical activity [247, 248] and several experimental
realizations of it have been reported most recently [85, 86].

In this section, we address the properties of strongly-repelling impenetrable
bosons in rotating ring-shaped or 2D harmonic traps. It has been found that
impenetrable bosons are “localized” relative to each other [60, 63, 85] and
exhibit nontrivial intrinsic crystalline correlations [60, 63]. For a small
number of bosons, $N$, these crystalline arrangements are reminiscent of the
structures exibited by the well-studied rotating electron molecules in quantum
dots under high magnetic fields [26, 52, 53]. Consequently, we use in the
following the term rotating boson molecules. A central result of our study is
that the point-group symmetries of the intrinsic crystalline structures give
rise to characteristic regular patterns (see below) in the ground-state
spectra and associated angular momenta of the RBMs as a function of the
rotational frequency for neutral bosons (or the magnetic field for charged
bosons).

An unexpected result of our studies is that the rotation of repelling bosons
(even those interacting weakly) does not necessarily lead to formation of
vortices, as is familiar from the case of rotating Bose-Einstein condenstates.
In particular, for small $N$, we will show that the Gross-Pitaevskii energies
(including those corresponding to formation of vortices) remain always higher
compared to the ground-state energies of the RBMs. Of course, we expect that
the rotating BEC will become the preferred ground state for sufficiently large
$N$ in the case of weakly repelling neutral bosons. We anticipate, however,
that it will be feasible to test our unexpected results for small $N$ by using
rotating optical lattices, where it is established that a small finite number
of atoms can be trapped per given site [87].

In a non-rotating trap, it is natural to describe a localized boson (at a
position ${\bf R}_{j}$) by a simple displaced Gaussian [60]. When the rotation
of the trap is considered, the Gaussian needs to be modified by a phase
factor, determined through the analogy between the one-boson Hamiltonian in
the rotating frame of reference and the planar motion of a charged particle
under the influence of a perpendicular magnetic field $B$ (described in the
symmetric gauge). That is, the single-particle wave function of a localized
boson is

$\varphi_{j}({\bf r})\equiv\varphi({\bf r},{\bf
R}_{j})=\frac{1}{\sqrt{\pi}\lambda}\exp\left[\frac{({\bf r}-{\bf
R}_{j})^{2}}{2\lambda^{2}}-\rmi{\bf r}{\bf\cdot}({\bf Q}\times{\bf
R}_{j})\right],$ (8.1)

with ${\bf Q}\equiv{\bf\hat{z}}/\Lambda^{2}$ and the width of the Gaussian
$\lambda$ is a variational parameter; $\Lambda\equiv
l_{B}\sqrt{2}=\sqrt{2\hbar c/(eB)}=\sqrt{2\hbar/(m\omega_{c})}$ for the case
of a perpendicular magnetic field ${\bf B}$, and $\Lambda\equiv
l_{\Omega}\sqrt{2}=\sqrt{\hbar/(m\Omega)}$ in the case of a rotating trap with
rotational frequency ${\bf\Omega}$ (we recall that $\omega_{c}\rightarrow
2\Omega$, see the Appendix). Note that we consider a 2D trap, so that ${\bf
r}\equiv(x,y)$ and ${\bf R}\equiv(X,Y)$.

Figure 27: (Color online) Properties of $N=8$ neutral repelling bosons in a
rotating toroidal trap as a function of the reduced rotational frequency
$\Omega/\omega_{0}$. The confining potential is given by (8.4) with $n=2$ and
radius $r_{0}=3l_{0}$, and the interaction-strength parameter was chosen as
$R_{\delta}=50$. (a) RBM ground-state energies, $E^{\rm{PRJ}}$. The inset
shows the range $0\leq\Omega/\omega_{0}\leq 0.3$. The numbers denote ground-
state magic angular momenta. (b) Energy difference
$E^{\rm{PRJ}}-E^{\rm{UBHF}}$. (c) Total angular momenta associated with (i)
the RBM ground states [thick solid line (showing steps and marked as PRJ);
online black] and (ii) the UBHF solutions (thin solid line; online red). In
the figures, we may use the symbol $L_{z}$, instead of simply $L$, to denote
the 2D total angular momentum.

The Hamiltonian corresponding to the single-particle kinetic energy is given
by

$H_{K}({\bf r})=({\bf p}-\hbar{\bf Q}\times{\bf r})^{2}/(2m),$ (8.2)

for the case of a magnetic field, and by

$H_{K}({\bf r})=({\bf p}-\hbar{\bf Q}\times{\bf r})^{2}/(2m)-m\Omega^{2}{\bf
r}^{2}/2,$ (8.3)

for the case of a rotating frame of reference.999The single-particle wave
function in (8.1) and the many-body projected wave function in (8.5) contain
contributions from higher Landau levels. These wave functions belong
exclusively to the lowest Landau level only in the limit when
$\lambda=\sqrt{2}l_{B}$ in the case of a magnetic field, or
$\lambda=\sqrt{2}l_{\Omega}$ and $\Omega/\omega_{0}=1$ in the case of a
rotating trap.

A toroidal trap with radius $r_{0}$ can be specified by the confining
potential

$V({\bf r})=\frac{\hbar\omega_{0}}{2}(r-r_{0})^{n}/l_{0}^{n},$ (8.4)

with $l_{0}=\sqrt{\hbar/(m\omega_{0})}$ being the characteristic length of the
2D trap. For $n\gg 2$ and $l_{0}/r_{0}\to 0$ this potential approaches the
limit of a toroidal trap with zero width, which has been considered often in
previous theoretical studies (see, e.g., Ref. [249]). In the following, we
consider the case with $n=2$, which is more realistic from the experimental
point of view. In this case, in the limit $r_{0}=0$, one recovers a harmonic
trapping potential.

To construct an RBM variational many-body wave function describing $N$
impenetrable bosons in the toroidal trap, we use $N$ displaced orbitals
$\varphi({\bf r},{\bf R}_{i})$, $i=1,2,\ldots,N$ [see (8.1)] centered at the
vertices of a regular polygon. Then, we first construct an unrestricted Bose
Hartree-Fock permanent [60, 63]
$|\Phi^{\rm{UBHF}}_{N}\rangle\propto\sum_{P(i_{m})}\varphi_{1}({\bf
r}_{i_{1}})\varphi_{2}({\bf r}_{i_{2}})\ldots\varphi_{N}({\bf r}_{i_{N}})$.
The UBHF permanent breaks the circular symmetry of the many-body Hamiltonian.
As was discussed in section 3.2, the “symmetry dilemma” is resolved through a
subsequent “symmetry-restoration” step accomplished via projection techniques
[23, 24, 30, 31, 52, 53], i.e., we construct a many-body wave function with
good total angular momentum by applying the projection operator $\hat{\cal
P}_{L}=(1/2\pi)\int_{0}^{2\pi}\rmd\theta\exp[\rmi\theta(L-\hat{L})]$, so that
the final RBM wave function is given by

$|\Psi^{\rm{PRJ}}_{N,L}\rangle=\frac{1}{2\pi}\int_{0}^{2\pi}\rmd\theta|\Phi^{\rm{UBHF}}_{N}(\theta)\rangle\rme^{\rmi\theta
L}.$ (8.5)

$|\Phi^{\rm{UBHF}}_{N}(\theta)\rangle$ is the original UBHF permanent rotated
by an azimuthal angle $\theta$. We note that, in addition to having good
angular momenta, the projected wave function $|\Psi^{\rm{PRJ}}_{N,L}\rangle$
has also a lower energy than that of $|\Phi^{\rm{UBHF}}_{N}\rangle$ [see, e.g.
$E_{L}^{\rm{PRJ}}-E^{\rm{UBHF}}$ in figure 27(b). The projected ground-state
energy is given by

$E_{L}^{\rm{PRJ}}=\int_{0}^{2\pi}h(\theta)\rme^{\rmi\theta
L}\rmd\theta\left/\int_{0}^{2\pi}n(\theta)\rme^{\rmi\theta
L}\right.\rmd\theta,$ (8.6)

where $h(\theta)=\langle\Phi_{N}^{\rm{UBHF}}(\theta=0)|{\cal
H}|\Phi_{N}^{\rm{UBHF}}(\theta)\rangle$ and
$n(\theta)=\langle\Phi_{N}^{\rm{UBHF}}(\theta=0)|\Phi_{N}^{\rm{UBHF}}(\theta)\rangle$;
the latter term ensures proper normalization.

Figure 28: (Color online) Single-particle densities and CPDs for $N=8$ bosons
in a rotating toroidal trap with $\Omega/\omega_{0}=0.2$ and $R_{\delta}=50$.
The remaining trap parameters are as in figure 27. (a) Gross-Pitaevskii
single-particle density. (b) UBHF single-particle density exhibiting breaking
of the circular symmetry. (c) RBM single-particle density exhibiting circular
symmetry. (d) CPD for the RBM wave function [PRJ wave function, see Equation
(8.5)] revealing the hidden point-group symmetry in the intrinsic frame of
reference. The observation point is denoted by a white dot. The RBM ground-
state angular momentum is $L_{z}=16$. Lengths in units of $l_{0}$. The
vertical scale is the same for (b), (c), and (d), but different for (a).

The many-body Hamiltonian in the rotating trap is given by

${\cal H}=\sum_{i=1}^{N}[H_{K}({\bf r}_{i})+V({\bf
r}_{i})]+\sum_{i<j}^{N}v({\bf r}_{i},{\bf r}_{j}),$ (8.7)

with the interparticle interaction being given by a contact potential
$v_{\delta}({\bf r}_{i},{\bf r}_{j})=g\delta({\bf r}_{i}-{\bf r}_{j})$ for
neutral bosons and a Coulomb potential $v_{C}({\bf r}_{i},{\bf
r}_{j})=Z^{2}e^{2}/|{\bf r}_{i}-{\bf r}_{j}|$ for charged bosons. The
parameter that controls the strength of the interparticle repulsion relative
to the zero-point kinetic energy is given by $R_{\delta}=gm/(2\pi\hbar^{2})$
[60, 63] for a contact potential and $R_{W}=Z^{2}e^{2}/(\hbar\omega_{0}l_{0})$
[20, 60] for a Coulomb repulsion.

For a given value of the dimensionless rotational frequency,
$\Omega/\omega_{0}$, the projection yields wave functions and energies for a
whole rotational band comprising many angular momenta. In the following, we
focus on the ground-state wave function (and corresponding angular momentum
and energy) associated with the lowest energy in the band.

Figure 29: (Color online) Properties of $N=6$ neutral bosons in a rotating
harmonic trap as a function of the reduced rotational frequency
$\Omega/\omega_{0}$. The confining potential is given by (8.4) with $n=2$ and
$r_{0}=0$, and the interaction-strength parameter was chosen as
$R_{\delta}=50$. The intrinsic molecular structure is $(1,5)$. (a) RBM ground-
state energies, $E^{\rm{PRJ}}$. The inset shows a smaller range. The numbers
denote ground-state angular momenta. (b) Total angular momenta associated with
(i) the RBM ground states (thick solid line showing steps; online black) and
(ii) the UBHF solutions (thin solid line; online red).

Figure 27(a) displays the ground-state energy $E_{\rm{PRJ}}$ of $N=8$ bosons
in a toroidal trap as a function of the dimensionless rotational frequency
$\Omega/\omega_{0}$, with $\omega_{0}$ being the trap frequency. The prominent
features in figure 27(a) are: (i) the energy diminishes as $\Omega/\omega_{0}$
increases; this is an effect of the centrifugal force, and (ii) the
$E_{\rm{PRJ}}$ curve consists of linear segments, each one associated with a
given angular momentum $L$. Most remarkable is the regular variation of the
values of $L$ with a constant step of $N$ units (here $N=8$) [see inset in
figure 27(a) and figure 27(c)]. These preferred angular momenta $L=kN$ with
integer $k$, are reminiscent of the so called “magic angular momenta” familiar
from studies of electrons under high-magnetic fields in 2D semiconductor
quantum dots [26, 52, 53].

Figure 30: (Color online) Properties of GP solutions (thin solid line; online
red) versus those of RBM wave functions (thick solid line; online green) for
$N=6$ neutral bosons as a function of the reduced rotational frequency
$\Omega/\omega_{0}$. A harmonic trap is considered, and the interaction
strength equals $R_{\delta}=50$. (a) Ground-state energies. (b) Associated
ground-state angular momenta. (c) GP (BEC) single-particle density at
$\Omega/\omega_{0}=0.65$ having 7 vortices with a 6-fold symmetry (thus
exhibiting breaking of the circular symmetry). (d) RBM single-particle density
at $\Omega/\omega_{0}=0.65$ which does not break the circular symmetry. (e)
CPD of the RBM at $\Omega/\omega_{0}=0.65$ revealing the intrinsic (1,5)
crystalline pattern. The white dot denotes the observation point ${\bf
r}_{0}$. Note the dramatic difference in spatial extent between the GP and RBM
wave functions [compare (c) with (d) and (e). Lengths in units of $l_{0}$. The
vertical scale is the same for (d) and (e), but different for (c).

The preferred angular momenta reflect the intrinsic molecular structure of the
localized impenetrable bosons. We note, that the (0,8) polygonal-ring
arrangement is obvious in the single-particle density associated with the UBHF
permanent [see figure 28(b)]; (0,8) denotes no particles in the inner ring and
8 particles in the outer one. After restoration of symmetry, however, the
single-particle density is circularly symmetric [see the PRJ single-particle
density in figure 28(c)] and the intrinsic crystallinity becomes “hidden”; it
can, however, be revealed via the conditional probability distribution [20,
52, 53, 60] [CPD, see figure 28(d)]. We note the Gross-Pitaevskii single-
particle density in figure 28(a), which is clearly different from the PRJ
density in figure 28(c).

The internal structure for charged bosons in a toroidal trap (not shown) is
similar to that of neutral bosons (figure 28), i.e., a (0,8) ring arrangement,
portrayed also in the stepwise variation (in steps of 8 units) of the total
angular momenta. The internal structure is also reflected in the variation of
the ground-state total energy as a function of the magnetic field. In contrast
to the case of neutral bosons, however, the ground-state energy curve for
charged bosons is not composed of linear segments, but of intersecting
inverted-parabola-type pieces; this is due to the positive contribution of the
Lorentz force compared to the negative contribution of the centrifugal force
in a rotating trap.

For RBMs in rotating harmonic traps, the polygonal-ring pattern of localized
bosons becomes more complex than the simple $(0,N)$ arrangement that appears
naturally in a toroidal trap. Indeed, in harmonic traps, one anticipates the
emergence of concentric ring structures. For $N=6$ neutral bosons in a
harmonic trap, we observe that, as in the case of a toroidal trap, the ground-
state energy as a function of the reduced rotational frequency,
$\Omega/\omega_{0}$, [figure 29(a)] is composed of linear segments, but now
the corresponding magic angular momenta [figure 29(b)] vary in steps of
$N-1=5$ units. This indicates a rotating boson molecule consisting of two
polygonal rings; denoted as a $(1,5)$ structure, with the inner ring having a
single boson and the outer ring five.

Figure 31: (Color online) Properties of GP solutions (thin solid line; online
red) versus those of RBM wave functions (thick solid line; online green) for
$N=6$ bosons as a function of the interaction strength $R_{\delta}$. A
harmonic trap is considered, and the reduced rotational frequency equals
$\Omega/\omega_{0}=0.85$. (a) Ground-state energies (b) Associated ground-
state angular momenta.

In figure 30(a), we display the rotating-boson-molecule and mean-field Gross-
Pitaevskii ground-state energies of $N=6$ strongly repelling (i.e.,
$R_{\delta}=50$) neutral bosons in a harmonic trap as a function of the
reduced angular frequency of the trap. The GP curve (thin solid line; online
red) remains well above the RBM curve (thick solid line; online green) in the
whole range $0\leq\Omega/\omega_{0}\leq 1$. The RBM ground-state angular
momenta exhibit again the periodicity in steps of five units [figure 30(b)].
As expected, the GP total angular momenta are quantized [$L_{z}=0$ (no-vortex)
or $L_{z}=6$ (one central vortex)] only for an initial range
$0\leq\Omega/\omega_{0}\leq 0.42$. For $\Omega/\omega_{0}\geq 0.42$, the GP
total angular momentum takes non-integer values and ceases to be a good
quantum number, reflecting the broken-symmetry character of the associated
mean field, with each kink signaling the appearance of a different vortex
pattern of $p$-fold symmetry ($p=1,2,3,4,\ldots$) [250]; see an example in
figure 30(c). The energetic superiority of the RBM wave function over the GP
solution demonstrated in figure 30(a) was to be expected, since we considered
the case of strongly repelling bosons. Unexpectedly, however, for a small
number of neutral bosons the energetic advantage of the rotating boson
molecule persists even for weakly repelling bosons, as illustrated in figure
31(a). Indeed, figure 31(a) displays the RBM (thick solid line; online green)
and GP (thin solid line; online red) ground-state energies for $N=6$ neutral
bosons in a trap rotating with $\Omega/\omega_{0}=0.85$ as a function of the
interaction parameter $R_{\delta}$. The surprising result in figure 31(a) is
that the GP energy remains above the RBM curve even for $R_{\delta}\rightarrow
0$. Of course the RBM wave function is very close to that of a BEC without
vortices when $R_{\delta}\rightarrow 0$ (BECs without vortices are
approximately feasible for small $N$). However, for small $N$, our results
show that BECs with vortices (i.e., for $L_{z}\geq N$) are not the preferred
many-body ground states; instead, formation of RBMs is favored. Note that the
energy difference $E^{\rm{GP}}-E^{\rm{PRJ}}$ increases rapidly with increasing
$R_{\delta}$, reflecting the fact that the RBM energies saturate (as is to be
expected from general arguments), while the GP energies (even with vortices
fully accounted for) exhibit an unphysical divergence as
$R_{\delta}\rightarrow\infty$ [figure 31(a)]; we have checked this trend up to
values of $R_{\delta}=100$ (not shown). Of interest again is the different
behavior of the RBM and GP ground state angular momenta [figure 31(b)] (see
also discussion of figure 30(b)).

To summarize this section: We have studied the ground-state properties of a
variational many-body wave function for repelling bosons in rotating traps
that incorporates correlations beyond the Gross-Pitaevskii mean-field
approximations. This variational wave function describes rotating boson
molecules, i.e., localized bosons arranged in polygonal-ring-type patterns in
their intrinsic frame of reference. For small numbers of neutral bosons, and
in particular in the case of GP vortex formation, the RBM ground-state
energies are lower than those associated with the corresponding Gross-
Pitaevskii BEC solutions. Given the large differences between the properties
of the RBM and BEC wave functions (which become more pronounced for larger
interaction parameter $R_{\delta}$), and the recently demonstrated ability to
experimentally control $R_{\delta}$ [85, 86, 87, 246], we anticipate that our
results could be tested in experiments involving rotating optical lattices.
Detection of rotating boson molecules could be based on a variety of
approaches, such as the measurement of the spatial extent [contrast the RBM
and BEC spatial extents in figure 30(c) - figure 30(e)], or the use of Hanbury
Brown-Twiss-type experiments [251] to directly detect the intrinsic
crystalline structure of the RBM.

###  8.2 Exact diagonalization for bosons in the lowest Landau level

Rotating ultracold trapped Bose condensed systems are most commonly discussed
in the context of formation of vortex lattices, which are solutions to the
Gross-Pitaevskii mean-field equation [4, 5, 252, 253, 254, 255, 256, 257].
Such vortex lattices have indeed been found experimentally for systems
containing a large number of bosons [258, 259, 260]. Nevertheless, several
theoretical investigations [67, 68, 69, 70, 71] of rapidly rotating trapped
bosonic systems suggested formation of strongly correlated exotic states which
differ drastically from the aforementioned vortex-lattice states. While
experimental realizations of such strongly correlated states have not been
reported yet, there is already a significant effort associated with two-
dimenional exact-diagonalization studies of a small number of particles ($N$)
in the lowest Landau level; the LLL restriction corresponds to the regime of
rapid rotation, where the rotational frequency of the trap $\Omega$ equals the
frequency of the confining potential. The large majority [67, 69, 70, 71] of
such exact-diagonalization studies have attempted to establish a close
connection between rapidly rotating bosonic gases and the physics of electrons
under fractional-quantum-Hall-effect conditions employing the bosonic version
of “quantum-liquid” analytic wave functions, such as the Laughlin wave
functions, composite-fermion, Moore-Read, and Pfaffian functions.

As described in section 6, the “quantum-liquid” picture for a small number of
trapped electrons in the FQHE regime has been challenged in a series of
extensive studies [24, 26, 42, 51, 52, 53] of electrons in 2D quantum dots
under high magnetic fields. Such studies (both exact-diagonalization and
variational) revealed that, at least for finite systems, the underlying
physical picture governing the behavior of strongly-correlated electrons is
not that of a “quantum liquid.” Instead, the appropriate description is in
terms of a “quantum crystal,” with the localized electrons arranged in
polygonal concentric rings [24, 26, 51, 52, 53, 127, 128, 131]. These
“crystalline” states lack [52, 53] the familiar rigidity of a classical
extended crystal, and are better described [24, 26, 42, 51, 52, 53] as
rotating electron (or Wigner) molecules.

Motivated by the discovery in the case of electrons of REMs at high $B$ (and
from the fact that Wigner molecules form also at zero magnetic field [20, 25,
41, 50, 167, 188]) some theoretical studies have most recently shown that
analogous molecular patterns of localized bosons do form in the case of a
small number of particles inside a static or rotating harmonic trap [43, 60,
63, 177, 178]. In analogy with the electron case, the bosonic molecular
structures can be referred to [63] as rotating boson molecules; a description
of RBMs via a variational wave function built from symmetry-breaking displaced
Gaussian orbitals with subsequent restoration of the rotational symmetry was
presented in Refs. [43, 60, 63] and reviewed in section 8.1.

In a recent paper, Baksmaty et al[72] used exact diagonalization in the lowest
Landau level to investigate the formation and properties of RBMs focusing on a
larger number of particles than previously studied, in particular for sizes
where multiple-ring formation can be expected based on our knowledge of the
case of 2D electrons in high $B$. A finite number of particles ($N\leq 11$) at
both low ($\nu<1/2$) and high ($\nu\geq 1/2$) filling fractions $\nu\equiv
N(N-1)/2L$ (where $L\equiv{\cal L}/\hbar$ is the quantum number associated
with the total angular momentum ${\cal L}$) was studied and both the cases of
a long-range (Coulomb) and a short-range ($\delta$-function) repulsive
interaction were investigated. In this section, we report some main results
from Ref. [72].

As in the case of electrons in 2D quantum dots, we probe the crystalline
nature of the bosonic ground states by calculating the full anisotropic two-
point correlation function $P({\bf r},{\bf r}_{0})$ [see Equation (1.1)]
associated with the exact wavefunction $\Psi({\bf r}_{1},{\bf
r}_{2},\ldots,{\bf r}_{N})$. The quantity $P({\bf r},{\bf r}_{0})$ is
proportional to the probability of finding a boson at ${\bf r}$ given that
there is another boson at the observation point ${\bf r}_{0}$, and it is often
referred to as the conditional probability distribution (section 1.5). A main
finding of our studies is that consideration solely of the CPDs is not
sufficient for the boson case at high fractional fillings $\nu\geq 1/2$; in
this case, one needs to calculate even higher-order correlation functions,
e.g., the full $N$-point correlation function defined as the modulus square of
the full many-body EXD wave function, i.e.,

$P({\bf r};{\bf r}_{1},{\bf r}_{2},\ldots,{\bf r}_{N-1})=|\Psi({\bf r};{\bf
r}_{1},{\bf r}_{2},\ldots,{\bf r}_{N-1})|^{2},$ (8.8)

where one fixes the positions of $N-1$ particles and inquiries about the
(conditional) probability of finding the $N$th particle at any position ${\bf
r}$.

The investigations in this section are also motivated by recent experimental
developments, e.g., the realization of trapped ultracold gas assemblies
featuring bosons interacting via a long-range dipole-dipole interaction [261,
262]. We expect the results presented in this section to be directly relevant
to systems with a two-body repulsion intermediate between the Coulomb and the
delta potentials. Additionally, we note the appearance of promising
experimental techniques for measuring higher-order correlations in ultra-cold
gases employing an atomic Hanbury Brown-Twiss scheme [251] or shot-noise
interferometry [263, 264]. Experimental realization of few-boson rotating
systems can be anticipated in the near future as a result of increasing
sophistication of experiments involving periodic optical lattices co-rotating
with the gas, which are capable of holding a few atoms in each site. A natural
first step in the study of such systems is the analysis of the physical
properties of a few particles confined in a rotating trap with open boundary
conditions (i.e., conservation of the total angular momentum $L$).

Figure 32: Ground-state angular momenta, $L_{\rm{gs}}$, for $N=6$ bosons in a
rapidly rotating trap [described by the LLL Hamiltonian in (8.9)], as a
function of the rotational frequency $\Omega$ expressed in units of
$\omega_{0}$. The bosons interact via a Coulombic repulsion (left) and a delta
repulsion (right), and the many-body Hilbert space is restricted to the lowest
Landau level. The angular momentum associated with the first bosonic Laughlin
state occurs at $L=30$, i.e., at $N(N-1)$. The value of
$c=0.2\hbar\omega_{0}\Lambda$ for the Coulomb case (left) and the value of
$g=2\pi\hbar\omega_{0}\Lambda^{2}/N$ for the case of a delta repulsion
(right); the many-body wave functions do not depend on these choices. In the
delta-interaction case, the values of the angular momenta terminate with the
value $L=30$ (the Laughlin value) at $\Omega/\omega_{0}=1$. In contrast, in
the Coulomb-interaction case (left), the values of the ground-state angular
momenta do not terminate, but diverge as $\Omega/\omega_{0}\rightarrow 1$.
Note the stepwise variation of the values of the ground-state angular momenta
in both cases, indicating the presence of an intrinsic point-group symmetry
associated with the (0,6) and (1,5) polygonal-ring structure of a rotating
Boson molecule.

The main results of Ref. [72] can be summarized as follows: Similar to the
well-established (see section 6 and section 7) emergence of rotating electron
molecules in quantum dots, rotating boson molecules form in rotating harmonic
traps as well. The RBMs are also organized in concentric polygonal rings that
rotate independently of each other, and the polygonal rings correspond to
classical equilibrium configurations and/or their low-energy isomers.
Furthermore, the degree of crystallinity increases gradually with larger
angular momenta $L$’s (smaller filling fractions $\nu$’s), as was the trend
[26, 52, 53] for the REMs and as was observed also for $\nu<1/2$ in another
study [178] for rotating bosons in the lowest Landau level with smaller $N$
and single-ring structures. We finally note that the crystalline character of
the RBMs appears to depend only weakly on the range of the repelling
interaction, for both the low (see also Ref. [178]) and high (unlike Ref.
[177]) fractional fillings.

In studies of 2D quantum dots, CPDs were used some time ago in Refs. [50, 128,
131]. For probing the intrinsic molecular structure in the case of ultracold
bosons in 2D traps, however, they were introduced only recently by Romanovsky
et al[60]. The importance of using CPDs as a probe can hardly be
underestimated. Indeed, while exact-diagonalization calculations for bosons in
the lowest Landau level have been reported earlier [67, 68, 69, 70, 71], the
analysis in these studies did not include calculations of the CPDs, and
consequently formation of rotating boson molecules was not recognized.

Figure 33: (a) Single-particle densities [$n({\bf r})$; left column], (b) CPDs
$\left[P({\bf r},{\bf r}_{0})\right]$ in 3D plots (middle column), and (c)
CPDs in contour plots (right column), portraying the strengthening of the
crystalline RBM structure for $N=6$ bosons interacting via a repulsive Coulomb
interaction as the filling fraction $\nu$ is reduced. The white dots in the
CPD plots indicate the reference point ${\bf r}_{0}$. We note in particular
the gradual enhancement of the peak at the center of the plots, and the growth
of the radius of the outer ring; the latter reflects the nonrigid-rotor nature
of the RBMs (in analogy with the findings of Ref. [52] regarding the
properties of rotating electron molecules). The cases of $\nu=1/4$ and
$\nu=1/8$ exhibit a clear $(1,5)$ crystalline arrangement, while the case of
$\nu=1/2$ (first Laughlin state) is intermediate between a $(1,5)$ and a
$(0,6)$ pattern (see text for details). Lengths in units of $\Lambda$. The
vertical scales are in arbitrary units, which however do not change for the
panels within the same column (a), (b), or (c).

####  8.2.1 The case of $N=6$ bosons in the lowest Landau level.

As a specific example of the points discussed above in section 8.2, we present
here results for $N=6$ bosons in the lowest Landau level. For additional cases
(e.g., $N=9$ and $N=11$), see Ref. [72].

In analogy with the magnetic-field Hamiltonian of Equation (7.1), the many-
body Hamiltonian for $N$ bosons in a rotating trap is reduced in the lowest
Landau level to the expression

$\tilde{\cal
H}^{\Omega}_{\rm{LLL}}=N\hbar\omega_{0}+\hbar(\omega_{0}-\Omega)L+\sum_{i<j}^{N}v({\bf
r}_{i},{\bf r}_{j}),$ (8.9)

where $\omega_{0}$ specifies the 2D harmonic trap and $\Omega$ denotes the
rotational frequency. The interparticle interaction is given by a contact
potential $v_{\delta}({\bf r}_{i},{\bf r}_{j})=g\delta({\bf r}_{i}-{\bf
r}_{j})$ for neutral bosons and a Coulomb potential $v_{C}({\bf r}_{i},{\bf
r}_{j})=c/|{\bf r}_{i}-{\bf r}_{j}|$ for charged bosons.

Since $\tilde{\cal H}^{\Omega}_{\rm{LLL}}$ is rotationally invariant, i.e.,
$[\tilde{\cal H}^{\Omega}_{\rm{LLL}},L]=0$, its eigenstates $\Psi_{L}$ must
also be eigenstates of the total angular momentum with eigenvalue $\hbar L$.
For a given rotational frequency $\Omega$, the eigenstate with lowest energy
is the ground state; we denote the corresponding angular momentum as
$L_{\rm{gs}}$.

We proceed to describe the EXD results for $N=6$ particles interacting via a
Coulomb repulsion by referring to figure 32, where we plot $L_{\rm{gs}}$
against the angular frequency $\Omega$ of the rotating trap. A main result
from all our calculations is that $L_{\rm{gs}}$ increases in characteristic
(larger than unity) steps that take only a few integer values, i.e., for $N=6$
the variations of $L_{\rm{gs}}$ are in steps of 5 or 6. In keeping with
previous work on electrons [24, 26, 42, 51, 52, 53] at high $B$, and very
recently on bosons in rotating traps [43, 60, 63, 177, 178], we explain these
magic-angular-momenta patterns (i.e., for $N=6$, $L_{\rm{gs}}=L_{0}+5k$ or
$L_{\rm{gs}}=L_{0}+6k$, with $L_{0}=0$) as manifestation of an intrinsic
point-group symmetry associated with the many-body wave function. This point-
group symmetry emerges from the formation of RBMs, i.e., from the localization
of the bosons at the vertices of concentric regular polygonal rings; it
dictates that the angular momentum of a purely rotational state can only take
values $L_{\rm{gs}}=L_{0}+\sum_{i}k_{i}n_{i}$, where $n_{i}$ is the number of
localized particles on the $i$th polygonal ring. [We remind the reader that
for spin-polarized electrons in the lowest Landau level, the corresponding
value is $L_{0}=N(N-1)/2$.] Thus for $N=6$ bosons, the series $L_{\rm{gs}}=5k$
is associated with an $(1,5)$ polygonal ring structure, while the series
$L_{\rm{gs}}=6k$ relates to an $(0,6)$ arrangement of particles. It is
interesting to note that in classical calculations [114] for $N=6$ particles
in a harmonic 2D trap, the $(1,5)$ arrangement is found to be the global
energy minimum, while the $(0,6)$ structure is the lowest metastable isomer.
This fact is apparently reflected in the smaller weight of the
$L_{\rm{gs}}=6k$ series compared to the $L_{\rm{gs}}=5k$ series, and the
gradual disappearance of the former with increasing $L$.

Magic values dominate also the ground state angular momenta of neutral bosons
(delta repulsion) in rotating traps, as shown for $N=6$ bosons in the right
panel of figure 32. Although the corresponding $\Omega$-ranges along the
horizontal axis may be different compared to the Coulomb case, the appearance
of only the two series $5k$ and $6k$ is remarkable — pointing to the formation
of RBMs with similar $(1,5)$ and $(0,6)$ structures in the case of a delta
interaction as well (see also Refs. [177, 178]). An important difference,
however, is that for the delta interaction both series end at
$\Omega/\omega_{0}=1$ with the value $L=N(N-1)=30$ (for $N=6$ the bosonic
Laughlin value at $\nu=1/2$), while for the Coulomb interaction this $L$ value
is reached for $\Omega/\omega_{0}<1$ — allowing for an infinite set of magic
angular momenta [larger than $N(N-1)$] to develop as
$\Omega/\omega_{0}\rightarrow 1$.

Beyond the analysis of the ground-state spectra as a function of $\Omega$, the
intrinsic crystalline point-group structure can be revealed by an inspection
of the CPDs [and to a much lesser extent by an inspection of single-particle
densities]. Because the EXD many-body wave function is an eigenstate of the
total angular momentum, the single-particle densities are circularly symmetric
and can only reveal the presence of concentric rings through oscillations in
the radial direction. The localization of bosons within the same ring can only
be revealed via the azimuthal variations of the anisotropic CPD [Equation
(1.1)]. One of our findings is that for a given $N$ the crystalline features
in the CPDs develop slowly as $L$ increases (or $\nu$ decreases).

For $\nu<1/2$, we find that the crystalline features are well developed for
all sizes studied by us. In figure 33, we present some concrete examples of
CPDs from exact-diagonalization calculations associated with the ground-states
of $N=6$ bosons in a rotating trap interacting via a repulsive Coulomb
potential. In particular, we present the CPDs for $L_{\rm{gs}}=30$ (bosonic
Laughlin for $\nu=1/2$), 60, and 120; these angular momenta are associated
with ground states at specific $\Omega$-ranges [see figure 32]. All three of
these angular momenta are divisible by both 5 and 6. However, only the
$L_{\rm{gs}}=30$ CPD (figure 33 top row) has a structure that is intermediate
between the $(1,5)$ and the $(0,6)$ polygonal-ring arrangements. The two other
CPDs, associated with the higher $L_{\rm{gs}}=60$ and $L_{\rm{gs}}=120$
exhibit clearly only the $(1,5)$ structure, illustrating our statement above
that the quantum-mechanical CPDs conform to the structure of the most stable
arrangement [i.e., the $(1,5)$ for $N=6$] of classical point-like charges as
the fractional filling decreases.

Figure 34: Contour plots of the CPD (a) and $N$-point correlation function (b)
and (c) for $N=6$ bosons with $L_{\rm{gs}}=15$ interacting via a Coulomb
repulsion. The white squares indicate the positions of the fixed particles.
The black square in (b) and (c) indicates the position of the 6th particle
according to the classical $(1,5)$ molecular configuration. Note the different
arrangements of the five fixed particles, i.e., (b) one fixed particle at the
center and (c) no fixed particle at the center. Note also that the CPD in (a)
fails to reveal the $(1,5)$ pattern, which, however, is clearly seen in the
$N$-point correlation functions in both (b) and (c). Lengths in units of
$\Lambda$. The vertical scales are arbitrary, but the same in (b) and (c).
Figure 35: Contour plots of the CPD (a) and $N$-point correlation function (b)
and (c) for $N=6$ bosons with $L_{\rm{gs}}=15$ interacting via a
$\delta$-repulsion. The white squares indicate the positions of the fixed
particles. The black square in (b) and (c) indicates the position of the 6th
particle according to the classical $(1,5)$ molecular configuration. Note the
different arrangements of the five fixed particles, i.e., (b) one fixed
particle at the center and (c) no fixed particle at the center. Note also that
the CPD in (a) fails to reveal the $(1,5)$ pattern, which, however, is clearly
seen in the $N$-point correlation functions in both (b) and (c). Lengths in
units of $\Lambda$. The vertical scales are arbitrary, but the same in (b) and
(c).

However, for $\nu>1/2$, the azimuthal variations may not be visible in the
CPDs, in spite of the characteristic step-like ground-state spectra [see
figure 32 for $N=6$ bosons]. This paradox is resolved when one considers
higher-order correlations, and in particular $N$-point correlations [see
Equation (8.8)]. In figure 34 and figure 35, we plot the $N$-point correlation
functions for $N=6$ bosons and $L_{\rm{gs}}=15$ for both the Coulomb
interaction and $\delta$-repulsion, respectively, and we compare them against
the corresponding CPDs. The value of 15 is divisible by 5, and one expects
this state to be associated with a $(1,5)$ molecular configuration. It is
apparent that the CPDs fail to portray such fivefold azimuthal pattern. The
$(1,5)$ pattern, however, is clear in the $N$-point correlations (middle and
right panels). One has two choices for choosing the positions of the first
five particles (white dots), i.e., one choice places one white dot at the
center and the other choice places all five white dots on the vertices of a
regular pentagon. For both choices, as shown by the contour lines in the
figures, the position of maximum probability for the sixth boson coincides
with the point that completes the $(1,5)$ configuration [see the black dots in
the middle and right panels].

Note that the differences in the CPDs and $N$-point correlation functions
between the Coulomb and the $\delta$-repulsion are rather minimal.

##  9 Summary

This report reviewed the physics of strong correlations in two-dimensional
small finite-size condensed-matter systems, such as electrons in quantum dots
and repelling bosons in harmonic traps. It was shown that strong correlations
in such systems relate to the appearance of symmetry breaking at the mean-
field level of description. Particular attention was given to the similarities
of symmetry breaking in these systems despite the different interparticle
interactions (Coulombic repulsion in quantum dots versus a contact potential
for neutral bososns in harmonic traps).

The universal aspects of symmetry breaking in small systems (including nuclei
and molecules in quantum chemistry) have been exploited to develop a two-step
method of symmetry breaking at the unrestricted Hartree-Fock level and
subsequent symmetry restoration via post Hartree-Fock projection techniques.
In conjunction with exact-diagonalization calculations, the two-step method
was used to describe a vast range of strongly-correlated phenomena associated
with particle localization and formation of crystalline (molecular) structures
of electrons in quantum dots and bosons in harmonic traps. Due to their finite
size, these crystalline structures are different from the familiar rigid
crystals of extended systems; they rather resemble and exhibit similarities
with the natural 3D molecules (e.g., ro-vibrational spectra).

It was shown that strongly-correlated phenomena emerging from symmetry
breaking include:

(I) Chemical bonding, dissociation, and entanglement in quantum dot molecules
and in electron molecular dimers formed within a single anisotropic quantum
dot, with potential technological applications to solid-state quantum-
computing devices.

(II) Electron crystallization, with localization on the vertices of concentric
polygonal rings, and formation of rotating electron molecules in circular
quantum dots. At zero magnetic field, the REMs can approach the limit of a
rigid rotor; at high magnetic field, the REMs are highly floppy, with the
rings rotating independently of each other.

(III) In the lowest Landau level, the rotating electron molecules are
described by parameter-free analytic many-body wave functions, which are an
alternative to the composite-fermion and Jastrow/Laughlin approaches, offering
a new point of view of the fractional quantum Hall regime in quantum dots
(with possible implications for the thermodynamic limit).

(IV) Crystalline phases of strongly repelling bosons. In the case of rotating
traps and in analogy with the REMs, such repelling bosons form rotating boson
molecules, which are energetically favored compared to the Gross-Pitaevkii
solutions, in particular in the regime of vortex formation.

This work is supported by the US D.O.E. (Grant No. FG05-86ER45234).
Calculations were peformed at the Georgia Institute of Technology Center for
Computational Materials Science.

## Appendix A

In this Appendix, we briefly review the single-particle wave functions and
associated energy spectra of a two-dimensional circular harmonic oscillator
under the influence of a perpendicular magnetic field $B$ (relevant to the
case of quantum dots) or under rotation with angular frequency $\Omega$
(relevant to the case of trapped atomic gases in rotating harmonic traps).
These single-particle wave functions and associated spectra are known as the
Darwin-Fock states and energy levels, after the names of the authors of two
original papers [162, 163] on this subject.

Figure 36: (Color online) The Darwin-Fock single-particle energy levels of a
2D harmonic oscillator under the effect of a perpendicular magnetic field $B$
as a function of $\eta=\omega_{c}/\omega_{0}$, where $\omega_{c}$ is the
cyclotron frequency and $\omega_{0}$ is the frequency specifying the 2D
harmonic confinement. A specific color (online) indicates orbitals with the
same number of radial nodes.

###  A.1  Two-dimensional isotropic oscillator in a perpendicular magnetic
field

In this case, the Hamiltonian (for an electron of mass $m^{*}$) is given by:

$H=\frac{1}{2m^{*}}({\bf p}-\frac{e}{c}{\bf
A})^{2}+\frac{1}{2}m^{*}\omega_{0}^{2}{\bf r}^{2},$ (1.1)

where ${\bf r}=(x,y)$ and $\omega_{0}$ is the frequency of the oscillator. In
the symmetric gauge, the vectror potential is given by ${\bf A}=({\bf
B}\times{\bf r})/2$, and the Hamiltonian (1.1) can be rewritten in the form

$H=\frac{{\bf
p}^{2}}{2m^{*}}-\frac{1}{2}\omega_{c}\hat{l}+\frac{1}{2}m^{*}\tilde{\omega}^{2}{\bf
r}^{2},$ (1.2)

where $\hat{l}=-\rmi\hbar(x\partial/\partial y-y\partial/\partial x)$ is the
angular momentum operator of the electron (in the $z$ direction),
$\omega_{c}=eB/(m^{*}c)$ is the cyclotron frequency, and
$\tilde{\omega}=\sqrt{\omega_{0}^{2}+\omega_{c}^{2}/4}$ is the effective-
confinement frequency.

The eigenfunctions of the Hamiltonian (1.2) have the same functional form as
those of a 2D harmonic oscillator at B=0, but with an effective frequency
$\tilde{\omega}$, i.e., in polar coordinates

$\phi_{n,l}(\rho,\theta)={\cal N}_{n,l}\rho^{|l|}\rme^{-\rho^{2}/2}\rme^{\rmi
l\theta}L_{n}^{|l|}(\rho^{2}),$ (1.3)

with $\rho=r/\tilde{l}$; the characteristic length $\tilde{l}$ is given by
$\tilde{l}=\sqrt{\hbar/(m^{*}\tilde{\omega})}$. In (1.3), the index $n$
denotes the number of nodes in the radial direction, and $l$ (without any
subscript or tilde) denotes the angular-momentum quantum numbers; the
$L_{n}^{|l|}$’s are associated Laguerre polynomials.

The single-particle energy spectrum corresponding to the Hamiltonian (1.2) is
plotted in figure 36; the associated eigenenergies are given by

$\frac{E_{n,l}}{\hbar\omega_{0}}=(2n+|l|+1)\sqrt{1+\frac{\eta^{2}}{4}}-\frac{l}{2}\eta,$
(1.4)

with $\eta=\omega_{c}/\omega_{0}$.

In the limit of $\omega_{c}/(2\omega_{0})\rightarrow\infty$, one can neglect
the external confinement, and the energy spectrum in Equation (1.4) reduces to
that of the celebrated Landau levels, i.e.,

$E_{\cal M}=\hbar\omega_{c}({\cal M}+\frac{1}{2}),$ (1.5)

where ${\cal M}=n+(|l|-l)/2$ is the Landau-level index.

We remark that the Landau levels are infinitely degenerate. The lowest Landau
level ${\cal M}=0$ contains all nodeless levels ($n=0$) with arbitrary
positive angular momentum $l\geq 0$.

Figure 37: (Color online) The Darwin-Fock single-particle energy levels of a
2D harmonic oscillator rotating with angular frequency $\Omega$ as a function
of $\eta=\Omega/\omega_{0}$, where $\omega_{0}$ is the frequency specifying
the 2D harmonic confinement. A specific color (online) indicates orbitals with
the same number of radial nodes.

###  A.2  Two-dimensional rotating harmonic oscillator

In the case of a rotating isotropic oscillator, instead of the expression
(1.2), one has the following single-particle Hamiltonian:

$H=\frac{{\bf p}^{2}}{2m}-\Omega\hat{l}+\frac{1}{2}m\omega_{0}^{2}{\bf
r}^{2},$ (1.6)

where the mass of the particle (e.g., a bosonic or fermionic atom) is denoted
by $m$; $\Omega$ denotes the rotational frequency.

From a comparison of the second terms in (1.2) and (1.6), one derives the
correspondence $\Omega\rightarrow\omega_{c}/2$.

We note that, unlike the application of a perpendicular magnetic field, the
rotation does not generate an effective confinement different from the
original external one [compare the third terms between (1.2) and (1.6)]. As a
result, the eigenfunctions of the Hamiltonian (1.6) are given by the
expressions (1.3), but with $\rho=r/l_{0}$ where the characteristic length
$l_{0}=\sqrt{\hbar/(m\omega_{0})}$.

The single-particle energy spectrum corresponding to the Hamiltonian (1.6) is
plotted in figure 37 and the associated eigenenergies are given by

$\frac{E_{n,l}}{\hbar\omega_{0}}=(2n+|l|+1)-l\eta,$ (1.7)

with $\eta=\Omega/\omega_{0}$.

For $\Omega/\omega_{0}=1$, the energy spectrum in Equation (1.7) reduces to
that of the corresponding Landau levels, i.e.,

$E_{\cal M}=2\hbar\omega_{0}({\cal M}+\frac{1}{2}),$ (1.8)

where ${\cal M}=n+(|l|-l)/2$ is the Landau-level index.

As was the case with the perpendicular magnetic field, the Landau levels are
infinitely degenerate, and the lowest Landau level ${\cal M}=0$ contains all
nodeless levels ($n=0$) with arbitrary positive angular momentum $l\geq 0$.
However, unlike the magnetic-field case where $\hbar\omega_{c}$ depends on
$B$, the energy gap between the Landau levels in the case of rotation is
independent of $\Omega$ and equals $2\hbar\omega_{0}$.

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