Oscillation bands of condensates on a ring: Beyond the mean field theory
C. G. Bao
Center of Theoretical Nuclear Physics,
National Laboratory of Heavy Ion Collisions,
Lanzhou 730000, P. R. China
The State Key Laboratory of Optoelectronic Materials and Technologies,
Zhongshan University, Guangzhou, 510275, P.R. China
Abstract: The Hamiltonian of a N-boson system confined on a ring with zero spin and repulsive
interaction is diagonalized. The excitation of a pair of p-wave-particles rotating reversely appears to
be a basic mode. The fluctuation of many of these excited pairs provides a mechanism of oscillation,
the states can be thereby classified into oscillation bands. The particle correlation is studied
intuitively via the two-body densities. Bose-clustering originating from the symmetrization of wave
functions is found, which leads to the appearance of 1-, 2-, and 3-cluster structures. The motion is
divided into being collective and relative, this leads to the establishment of a relation between the
very high vortex states and the low-lying states.
After the experimental realization of the Bose-Einstein
condensation1, various condensates confined under dif-
ferent circumstances have been extensively studied the-
oretically and experimentally. Mostly, the condensates
are considered to be confined in a harmonic trap. Con-
densates trapped by periodic potential have also been
studied due to the appearance of optical lattices.2 It
is believed that the appearance of condensates confined
in particular geometries is possible. Experimentally, the
particle interactions can now be tuned from very weak
to very strong,3−8 it implies that the particle correlation
may become important. Theoretically, to respond, go-
ing beyond the mean field Gross-Pitaevskii (GP) theory
is desirable, and the condensates confined in particular
geometries are also deserved to be considered.
Along this line, in addition to the ground state, the
yrast states have been studied both analytically and
numerically.9−17 The condensation on a ring has also
been studied recently.12 The present paper is also ded-
icated to the N−boson systems confined on a ring with
weak interaction, its scope is broader and covers the
whole low-lying spectra. A similar system has been in-
vestigated analytically by Lieb and Liniger16,17. How-
ever, the emphasis of their papers is different from the
present one, which is placed on analyzing the structures
of the excited states to find out their distinctions and sim-
ilarities, and to find out the modes of excitation. Based
on the analysis, an effort is made to classify the ex-
cited states. Traditionally, the particle correlation and
its effect on the geometry of N−boson systems is a topic
scarcely studied if N is large. In this paper, the corre-
lation is studied intuitively so as the geometric features
inherent in the excited states can be understood. Tradi-
tionally, a separation between the collective and internal
motions is seldom to be considered if N is large. In this
paper such a separation is made and leads to the estab-
lishment of a relation between the vortex states and the
low-lying states.
It is assumed that the N identical bosons confined on
a ring have mass m, spin zero, and square-barrier inter-
action. The ring has a radius R, N is given at 100, 20
and 10000. Let G = ~2/(2mR2) be the unit of energy.
The Hamiltonian then reads
H = −
i<j Vij (1)
where θi is the azimuthal angle of the i-th boson. Vij =
Vo if |θj − θi| ≤ θrange , or = 0 otherwise. Let φk =
eikθ/
2π be a single particle state, −kmax ≤ k ≤ kmax
is assumed. The N−body normalized basis functions
in Fock-representation are |α〉 ≡ |n−kmax , · · · nkmax〉,
where nj is the number of bosons in φj ,
nj = N,
njj = L, the total angular momentum. Then,
H is diagonalized in the space spanned by |α〉, the low-
lying spectrum together with the eigen-wave-functions,
each is a linear combination of |α〉, are thereby obtained.
Let Kα =
2 be the total kinetic energy of an |α〉
state. Evidently, those |α〉 with a large Kα are negligi-
ble for low-lying states. Therefore, one more constraint
Kα ≤ Kmax is further added to control the number of
|α〉. In this procedure, the crucial point is the calcula-
tion of the matrix elements of H . This can be realized
by using the fractional parentage coefficients18 (refer to
eq.(6) below). Numerical results are reported as follows.
This paper concerns only the cases with weak interac-
tion. Firstly, let Vo = 1, θrange = 0.025, and N = 100.
This is corresponding to γ = 0.00157 , where γ is in-
troduced by Lieb and Liniger to measure the strength
of interaction,16,17 this is shown later. When kmax and
Kmax are given at a number of values, the associated
eigen-energiesEj of the first, fifteenth, and sixteenth L =
0 eigen-states are listed in Table I. When (kmax,Kmax)
is changed from (3, 50) to (5,60), the total number of |α〉
is changed from 2167 to 8890. Table I demonstrates that
the great increase of basis functions does not lead to a
remarkable decrease of eigen-energies. Thus the conver-
gency is qualitatively satisfying even for the higher states.
http://arxiv.org/abs/0704.0842v1
TABLE I: Eigen-energies Ej (the unit is G) of the L = 0
states. The first row is (kmax,Kmax), the first column is the
serial number of states j. Vo = 1, θrange = 0.025, and N =
100 are given.
(3,50) (4,50) (4,60) (5,60)
1 39.109 39.090 39.090 39.078
15 53.645 53.616 53.616 53.613
16 54.822 54.800 54.800 54.790
In the following the choice kmax = 4 and Kmax = 50
are adopted, this limitation leads to a 3254-dimensional
space. Thereby the resultant data have at least three
effective figures, this is sufficient for our qualitative pur-
pose.
N=100,  L=0
FIG. 1: The spectrum of L = 0 states, the unit of energy is
G = ~2/(2mR2). N = 100, Vo = 1, and θrange = 0.025 are
assumed, they are the same for Fig.1 to Fig.5. The levels in a
column constitute an oscillation band, the levels in bold line
are doubly degenerate.
The low-lying spectrum is given in Fig.1, where the
lowest fourteen levels are included. Twelve of them can
be ascribed into three bands, in each band the levels
are distributed equidistantly, this is a strong signal of
harmonic-like oscillations. From now on the labels Ψ
and E(L,Z,i) are used to denote the wave function and en-
ergy of the i-th state of the Z-th band (Z=I, II, III,· · ·).
It turns out that the excitation of a pair of particles
both in p-wave but rotating reversely, namely, one par-
ticle in φ1 while the other one in φ−1, is a basic mode,
the pair is called a basic pair in the follows. A number of
such basic pairs might be excited. When 2j particles are
in basic pairs while the remaining N − 2j particles are
in φ0, the associated |α〉 is written as |P (j)〉. For all the
states of the I−band, we found Ψ(0)I,i is mainly a linear
combination of |P (j)〉 together with a small component
denoted by ∆I,i, i.e.,
I,i =
(0,I,i)
(j)〉+∆I,i (2)
where ∆I,i is very small as shown in Table II, while the
coefficients C
(L,Z,i)
j arise from the diagonalization. Thus
the basic structure of the I−band is just a fluctuation of
many of the basic pairs.
TABLE II: The weights of ∆Z,i of the bands with L = 0
i I−band II−band III−band
1 0.009 0.017 0.040
2 0.012 0.030 0.056
3 0.021 0.061 0.088
4 0.035 0.028
5 0.055 0.035
6 0.079 0.106
For lower states, C
(0,Z,i)
j would be very small if j
is larger, e.g., for the ground state, C
(0,I,1)
0 = 0.968
and C
(0,I,1)
j ≥2 ≈ 0, it implies that the excitation of many
pairs is not probable. It also implies that the ground state
wave function obtained via mean-field theory might be a
good approximation. However, for higher states, many
pairs would be excited. E.g., for the third state of the
I−band, C(0,I,3)j = 0.052, 0.406, 0.681,−0.528, and 0.245
when j is from 0 to 4, it implies a stronger fluctuation.
When a |α〉 has not only 2j particles in the basic pairs,
but also m particles in φk, while the remaining particles
in φ0, then it is denoted as |(k)mP (j)〉 (where k = ±1
are allowed) Similarly, we can define |(k1)m1(k2)m2P (j)〉,
and so on. For all the states of the II−band, we found
II,i =
(0,II,i)
[ |(2)1(−1)2P (j)〉
±|(−2)1(1)2P (j)〉] + ∆II,i
where both the + and − signs lead to the same energy,
thus the level is two-fold degenerate. Again, all the ∆II,i
are very small as shown in Table II, thus the fluctua-
tion of basic pairs is again the basic structure. However,
the II−band is characterized by having the additional
3-particle-excitation (one in d-wave and two in p-wave).
For all the states of the third band, we found
III,i
(0,III,i)
j |(2)1(−2)1P (j)〉+∆III,i (4)
Thus, the III−band contains, in addition to the fluc-
tuation of basic pairs, a more energetic pair with each
particle in d-wave. It was found that the spacing
E(0,Z,i+1) − E(0,Z,i) inside all the bands are nearly the
same, they are ∼3.15. This arises because they have the
same mechanism of oscillation, namely, the fluctuation of
basic pairs.
When the energy goes higher, more oscillation bands
can be found. The two extra levels in Fig.1 at the right
are the band-heads of higher bands.
Incidentally, the band-heads of the above three
bands are dominated by |P (0)〉, |(2)1(−1)2P (0)〉 ±
|(−2)1(1)2P (0)〉 and |(2)1(−2)1P (0)〉, respectively, and
their kinetic energies Kα = 0, 6, and 8. Among all the
basis functions with L = 0 and without basic pairs, these
three are the lowest three. This explains why the band-
heads are dominated by them. Once a band-head is fixed,
the corresponding oscillation band would grow up via the
fluctuation of basic pairs.
The particle correlations can be seen intuitively by ob-
serving the two-body densities
ρ2(θ1, θ2) =
dθ3 · · · dθN Ψ(L)∗Z,i Ψ
Similar to the calculation of the matrix elements of
interaction, the above integration can be performed in
coordinate space by extracting the particles 1 and 2
from |α〉 by using the fractional parentage coefficients18,
namely,
|α〉 =
nk(nk − 1)/N(N − 1)φk(1)φk(2)|αk〉
ka,kb
(ka 6=kb)
nkankb/N(N − 1)φka(1)φkb(2)|αkakb〉
where |αk〉 is different from |α〉 by replacing nk with nk−
2, |αkakb〉 is different from |α〉 by replacing nka and nkb
with nka − 1 and nkb − 1, respectively.
0 60 120 180 240 300 360
3 (c)
III-band
3 (b)
II-band
I-band
FIG. 2: ρ2 as functions of θ2 for the I ( a), II ( b), and III (c)
bands of L = 0 states, θ1 = 0 is given. The labels i of the
states Ψ
are marked by the curves.
ρ2 gives the spatial correlation between any pair of
particles as shown in Fig.2. For the ground state Ψ
ρ2 is flat implying that the correlation is weak. However,
it is a little larger when the two particles are opposite to
each other (θ1 = 0 and θ2 = π). It implies the existence
of a weak correlation which is entirely ignored by the
mean field theory. Thus, even the interaction adopted is
weak and even for the ground state, there is still a small
revision to the mean field theory. For higher states of the
I−band, the fluctuation of basic pairs becomes stronger.
Due to the fluctuation, the particles tend to be close to
each other to form a single cluster. This tendency is
clearly shown in Fig.2a.
For the first state of the III − band, Ψ(0)
III,1 has two
peaks in ρ2 implying a 2-cluster structure. It arises from
the two d-wave paticles inherent in the band. The fea-
ture of Ψ
II,1 is lying between Ψ
I,1 and Ψ
III,1. For all
higher states of every band, due to the strong fluctuation
of basic pairs, all the particles tend to be close to each
other as shown in 2b and 2c.
To understand the physics why the particles tend to be
close to each other, let us study the most important basis
state |P (j)〉. By inserting |P (j)〉 into eq.(5) to replace
and by using (6), ρ2 reads
ρ2(θ1, θ2) =
(2π)2N(N−1) [N(N − 1)− j(4N − 6j)
+4j(N − 2j)(1 + cos(θ1 − θ2)) + 4j2 cos2(θ1 − θ2)]
Where there are four terms at the right, the non-
uniformity arises from the third and fourth terms. The
third term causes the particles to be close to each other
to form a single cluster, while the fourth term causes
the two-cluster clustering. When j is small, the fourth
term can be neglected, and the particles tend to form
a single cluster. However, when j ≈ N/2 , the third
term can be neglected, and the particles tend to form
two clusters. It is noted that, if the symmetrization were
dropped, the density contributed by |P (j)〉 would be uni-
form. The appearance of the clustering originates from
the symmetrization of the bosonic wave functions, there-
fore it can be called as bose-clustering.
For L = 1 states, the lowest energy E(1,I,1) is higher
than E(0,I,1) by 1.606, but lower than E(0,I,2). Thus Ψ
is the true first excited state of the system. A number
of oscillation bands exist as well, the wave functions of
the lowest six bands are found as
(1,I,i)
j |(1)1P (j)〉+∆I,i
(1,II,i)
j |(2)1(−1)1P (j)〉+∆II,i
III,i
(1,III,i)
j |(−2)1(1)3P (j)〉+∆III,i
IV,i =
(1,IV,i)
j |(2)1(−2)1(1)1P (j)〉+∆IV,i
V,i =
(1,V,i)
j |(3)1(−1)2P (j)〉+∆V,i
V I,i =
(1,V I,i)
j |(3)1(−2)1P (j)〉+∆V I,i
Where the weights of all the ∆Z,i ≤ 0.1 if i ≤ 4. Thus,
just as the above L = 0 case, all the bands have the
common fluctuation of basic pairs, but each band has
a specific additional few-particle excitation. The ener-
gies of the band-heads from I to V I are 40.70, 45.42,
48.58, 50.14, 52.04, and 53.58 respectively. Further-
more, the spacing ∼3.15 found above is found again for
all these bands due to having the same mechanism of os-
cillation. The I−band is similar to the above I−band
with L = 0 but having an additional single p-wave ex-
citation, the ρ2 of them are one-one similar. Similarly,
the ρ2 of the IV−band is one-one similar to those of
the above III− band with L = 0. The ρ2 of the II
and III−bands are both similar to those of the above
II−band with L = 0. However, the V and V I−bands
are special due to containing the f-wave excitation, the ρ2
of their band-heads exhibit a 3-cluster structure as shown
in Fig.3. When the energy goes even higher, more higher
oscillation bands will appear. For the above six L = 1
bands, their band-heads are dominated by the |α〉 with
Kα = 1, 5, 7, 9, 11, and 13. Obviously, a higher Kα leads
to a higher band.
0 60 120 180 240 300 360
FIG. 3: ρ2 for selected L = 1 states. θ1 = 0, the (Z, i) labels
are marked by the curves.
In general, all the low-lying states can be classified into
oscillation bands. For all the lower bands disregarding
L, it was found that each band-head is dominated by
a basis function containing a specific few-particle exci-
tation but not containing any basic pairs. The energy
order of the bands is determined by the magnitudes of
Kα associated with the dominant basis function |α〉 of
the band-heads. Once a band-head stands, an oscillation
band will grow up from the band-head simply via the fluc-
tuation of basic pairs. For examples, for L = 2 states, the
dominant |α〉 of the band-heads of the four lowest oscil-
lation bands are |(1)2P (0)〉, |(2)1P (0)〉, |(−2)1(1)4P (0)〉,
and |(3)1(−1)1P (0)〉 with Kα = 2, 4, 8, and 10, respec-
tively.
For L = 3 states, the dominant |α〉 of the band-heads of
the three lowest bands are |(1)3P (0)〉, |(2)1(1)1P (0)〉, and
|(3)1P (0)〉, with Kα = 3, 5, and 9, respectively. Since the
p-, d-, and f-wave appear successively, these band-heads
exhibit 1-cluster, 2-cluster, and 3-cluster structures, re-
spectively, as shown in Fig.4.
0 60 120 180 240 300 360
III,1ρ
FIG. 4: ρ2 for the band-heads of L = 3 states, θ1 = 0.
Furthermore, a −L state can be derived from the
corresponding L state simply by changing every k to
−k, i.e., change the components |(k1)m1(k2)m2P (j)〉 to
|(−k1)m1(−k2)m2P (j)〉, and so on. Therefore Ψ(−L)Z,i =
)∗, and E(−L,Z,i) = E(L,Z,i).
0 2 4 6 8
Yrast line
N=100,   Vo=1,  θrange=0.025
FIG. 5: Energies of the yrast states with L = 0 to 10.
Let us study the yrast states Ψ
I,1 , each is the lowest
one for a given L. The energies of them are plotted in
Fig.5, their wave functions are found as
I,1 =
(L,I,1)
j |(1)LP (j)〉+∆LI,1 (9)
where ∆LI,i is very small. When L is small, the fluc-
tuation of basic pairs is small, and the yrast states are
dominated by the j = 0 component |(1)LP (0)〉.When L is
larger, the weight of the |(1)LP (0)〉 component becomes
smaller. E.g., when L = 0, 2, 4, and 10, the weights
of |(1)LP (0)〉 are 0.94, 0.84, 0.75, and 0.54, respectively.
Evidently, the energy going up linearly in the yrast line
in Fig.5 is mainly due to the linear increase of the number
of p-wave particles.
N=20,   L=0
III IV V
FIG. 6: The spectrum of the L = 0 states with N = 20,
Vo = 1, and θrange = 0.025. Refer to Fig.1
When N = 20, all the above qualitative features re-
main unchanged. Examples are given in Fig.6 and 7 to
be compared with Fig.1 and 4. Nonetheless, the decrease
of N implies that the particles have a less chance to meet
each other, thus the particle correlation is expected to be
weaker. Quantitatively, it was found that (i) The spac-
ing of adjacent oscillation levels becomes smaller, it is
now ∼2.2 to replace the previous 3.15 (ii) The fluctu-
ation becomes weaker. E.g., the weights of |P (j)〉 of
the Ψ
I,3 state are 0.01, 0.95, and 0.03 for j = 1, 2, and
3, respectively, while these weights would be 0.16, 0.46,
0 60 120 180 240 300 360
0.20 N=20,  L=3 
III,1
FIG. 7: The same as Fig.4 but with N = 20
and 0.28 if N = 100. (iii) When N becomes small, the
geometric features would become explicit. E.g., for the
3-cluster structure, the difference between the maximum
and minimum of ρ2 is ∼0.007 in Fig.4, but ∼ 0.033 in
Fig.7.
The decrease of Vo or θrange was found to cause an
effect similar to the decrease of N , the spectra would
remain qualitatively unchanged. Quantitatively, when
Vo is changed from 1 to 0.1, the spacing inside a band
is changed from ∼ 3.15 to ∼ 2.14, and the fluctuation
becomes much weaker as expected.
In what follows we study the vortex states. For an
arbitrary Lo ≤ N/2, the spectra of the Ψ(N−Lo)Z,i and
Z,i states are found to be identical
15, except the former
shifts upward as a whole by N − 2Lo, namely,
E(N−Lo,Z,i) = E(Lo,Z,i) +N − 2Lo (10)
Furthermore, their ρ2 are found to be identical.
Let us define an operator
X so that the state
X |α〉 is
related to |α〉 by changing every ki in |α〉 to −ki + 1,
i.e., φki(θ) to φ−ki+1(θ) = e
iθφ−ki(θ). We further found
from the numerical data that
(N−Lo)
Z,i =
holds exactly. In fact,
X causes a reversion of rota-
tion of each particle plus a collective excitation. It does
not cause any change in particle correlation, therefore ρ2
remains exactly unchanged. Thus the L large states,
including the vortex states L = N , can be known from
the L small states.
The underlying physics of this finding is the separabil-
ity of the Hamiltonian (it is emphasized that the sepa-
rability is exact as can be proved by using mathematical
induction). Let θcoll =
θi/N, which describes a col-
lective rotation. Then H = − 1
+Hint, where Hint
describes the relative (internal) motions and does not de-
pend on θcoll. Accordingly, E
(L,Z,i) = L2/N + E
(L,Z,i)
int ,
the former is for collective and the latter is for relative
(internal) motions. The eigen-states can be thereby sep-
arated as Ψ
Z,i =
eiLθcoll ψ
(L,Z,i)
int . The feature of the
internal states ψ
(L,Z,i)
int has been studied in [19]. Where
it was found that, for an arbitrary Lo
(N+Lo,Z,i)
int = ψ
(Lo,Z,i)
With these in mind, eq.(10) and (11) can be derived as
follows.
From the separability
(N−L)
ei(N−L)θcoll ψ
(N−L,Z,i)
int (13)
X acts on a wave function with L, from the defini-
tion of
X, L should be changed to −L and an additional
factor
eiθj = eiN θcoll should be added, thus
Z,i =
ei(N−L)θcoll ψ
(−L,Z,i)
Due to (12), the right hand sides of (13) and (14) are
equal, thereby (11) is proved.
Furthermore, since ψ
(−L,Z,i)
int = (ψ
(L,Z,i)
int )
∗, the internal
energy E
(N−L,Z,i)
int = E
(−L,Z,i)
int = E
(L,Z,i)
int . Therefore,
E(N−L,Z,i) − E(L,Z,i) = (N − L)2/N − L2/N = N − 2L.
This recovers eq.(10), the energy difference arises purely
from the difference in collective rotation.
If the particles are tightly confined on the ring, rapidly
rotating state with a large L = JN − Lo would exist,
where J is an integer. Their spectra would remain the
same but shift upward by J(JN − 2Lo) from the spec-
trum with L = Lo, while Ψ
(JN−Lo)
Z,i =
Z,i , where
XJ changes each φki to φ−ki+J . Thus the rapidly rotat-
ing states have the same internal structure as the corre-
sponding lower states but have a much stronger collective
rotation.
When N increases greatly while Vo or θrange decreases
accordingly, the qualitative behaviors remain unchanged.
E.g., when N = 10000 and Vo = 0.01 (θrange remains
unchanged), the spectrum and the wave functions are
found to be nearly the same as the case N = 100 and
Vo = 1, except that the spectrum has shifted upward
nearly as a whole by 3939. This is again a signal that,
for weak interaction and for the ground states, the mean-
field theory is a good approximation.
It is noted that the confinement by a ring is quite differ-
ent from a 2-dimensional harmonic trap. In the latter,
the energy of a particle in the lowest Landau levels is
proportional to its angular momentum k. However, for
the rings, it is proportional to k2. Consequently, higher
partial waves are seriously suppressed and the p-wave ex-
citation becomes dominant. For a harmonic trap it was
found in [10,11] that d- and f-wave excitations are more
important than the p-wave excitation when L is small.
This situation does not appear in our case.
When the zero-range interaction Vij = gδ(θi − θj) is
adopted, The results are nearly the same with those from
the square-barrier interaction if the parameters are re-
lated as g = 2Voθrange (in this choice both interactions
have the same diagonal matrix elements). For an exam-
ple, a comparison is made in Table III. The high simi-
larity between the two sets of data imply that the above
findings are also valid for zero-range interaction.
TABLE III: Eigen-energies of the four lowest L = 0 states
for a system with N = 100 and with zero-range interaction
Vij = 0.05δ(θi − θj) (the unit of energy is G as before). The
weights of the j = 0 components of these states are also
listed. The corresponding results from square-barrier inter-
action with Vo = 1, and θrange = 0.025 are given in the
parentheses.
(L,Z, i) E(L,Z,i) (C
(L,Z,i)
(0, I, 1) 39.0900 (39.0902) 0.9370 (0.9371)
(0, I, 2) 42.2982 (42.2983) 0.0510 (0.0510)
(0, I, 3) 45.4733 (45.4733) <0.02 (<0.02)
(0, II, 1) 47.0076 (47.0074) 0.8372 (0.8373)
The numerical results from using zero-range interac-
tion can be compared with the exact results from solving
integral equations by Lieb and Liniger [16,17]. The vari-
ables γ and e(γ) introduced in [16] are related to those of
this paper as γ = gπ/N and e(γ) = 4π2E/N3 (the unit
of E is G). However, this paper concerns mainly the case
of weak interaction, say, g ≤ 0.05,or γ ≤ 0.00157 (other-
wise, the procedure of diagonalization would not be valid
due to the cutoff of the space). Nonetheless, even γ is
as large as 0.5 (g = 15.9) the evolution of the ground
state energy with N = 100 against γ obtained via di-
agonalization coincide, in the qualitative sense, with the
exact results quite well . This is shown in Fig.8 to be
compared with Fig.3 of [16], where γ is ranged from 0
to 10. In Fig.8, the constraint γ < e(γ) is recovered.
Furthermore, when γ is small, e(γ) against γ appears as
a straight line.
0.0 0.1 0.2 0.3 0.4 0.5
FIG. 8: e(γ) = 4π2E/N3 against γ = gπ/N . N is given
at 100 and E is the ground state energy calculated from the
diagonalization in the unit G.
In summary, a detailed analysis based on the numerical
data of N−boson systems on a ring with weak interac-
tion has been made. The main result is the discovery of
the basic pairs, which exist extensively in all the excited
states and dominates the low-lying spectra. The fluc-
tuation of basic pairs provides a common mechanism of
oscillation, the low-lying states are thereby classified into
oscillation bands. Each band is characterized by having
its specific additional excitation of a few particles. Since
the mechanism of oscillation is common, the level spac-
ings of different bands are nearly equal in a spectrum.
To divide the motion into being collective and relative
provides a better understanding to the relation between
the higher and lower states. The very high vortex states
with L ≈ N can be understood from the correspond-
ing low-lying states because they have exactly the same
internal states.
The particle correlation has been intuitively studied.
particle densities are found to be in general non-uniform,
bose-clustering originating from the symmetrization of
wave functions is found, which leads to the appearance
of one, two, and three clusters. This phenomenon would
become explicit and might be observed if N is small.
Acknowledgment: The support by NSFC under the
grants 10574163 and 90306016 is appreciated.
REFERENCES
1, D.M. Stamper-Kurn, M.R. Andrews, A.P.
Chikkatur, S. Inouye, H.-J. Miesner, J. Stenger, and W.
Ketterle, Phys. Rev. Lett. 80, 2027 (1998)
2, B.P. Anderson and M.A. Kasevich, Science 281,
1686 (1998)
3, J.L. Roberts, et al, Phys. Rev. Lett. 81, 5159
(1998)
4, J. Stenger, et al, Phys. Rev. Lett. 82, 2422 (1999).
5, S.L. Cornish et al., Phys. Rev. Lett. 85, 1795 (2000)
6, M. Greiner et al., Nature (London) 415, 39 (2002)
7, B. Paredes et al., Nature (London) 429, 277 (2004)
8, G.T. Kinoshita, T. Wenger, and D.S. Weiss, Science
305, 1125 (2004)
9, N.K. Wilkin, J.M.F. Gunn, and R.A. Smith, Phys.
Rev. Lett. 80, 2265 (1998)
10, B. Mottelson, Phys. Rev. Lett. 83, 2695 (1999)
11, G.F. Bertsch and T. Papenbrock, 83, 5412 (1999)
12, K. Sakmann, A.I. Streltsov, O.E. Alon, L.S. Ceder-
baum, Phys. Rev. A 72, 033613 (2005)
13, I. Romanovsky, C. Yannouleas, and U. Landman,
Phys. Rev. Lett. 93, 230405 (2004)
14, I. Romanovsky, C. Yannouleas, L.O. Baksmaty,
and U. Landman, Phys. Rev. Lett. 97, 090401 (2006)
15, Yongle Yu, cond-mat/0609711 v1.
16, E.H. Lieb and W.Liniger, Phys, Rev. 130, 1605
(1963)
17, E.H. Lieb, Phys, Rev. 130, 1616 (1963)
http://arxiv.org/abs/cond-mat/0609711
18, F. Bacher and S. Goudsmit, Phys. Rev., 46, 948
(1934)
19, C.G. Bao, G.M. Huang, and Y.M. Liu, Phys. Rev.
B 72, 195310 (2005)

  The Hamiltonian of a N-boson system confined on a ring with zero spin and
repulsive interaction is diagonalized. The excitation of a pair of
p-wave-particles rotating reversely appears to be a basic mode. The fluctuation
of many of these excited pairs provides a mechanism of oscillation, the states
can be thereby classified into oscillation bands. The particle correlation is
studied intuitively via the two-body densities. Bose-clustering originating
from the symmetrization of wave functions is found, which leads to the
appearance of 1-, 2-, and 3-cluster structures. The motion is divided into
being collective and relative, this leads to the establishment of a relation
between the very high vortex states and the low-lying states.

Oscillation bands of condensates on a ring: Beyond the mean field theory
C. G. Bao
Center of Theoretical Nuclear Physics,
National Laboratory of Heavy Ion Collisions,
Lanzhou 730000, P. R. China
The State Key Laboratory of Optoelectronic Materials and Technologies,
Zhongshan University, Guangzhou, 510275, P.R. China
Abstract: The Hamiltonian of a N-boson system confined on a ring with zero spin and repulsive
interaction is diagonalized. The excitation of a pair of p-wave-particles rotating reversely appears to
be a basic mode. The fluctuation of many of these excited pairs provides a mechanism of oscillation,
the states can be thereby classified into oscillation bands. The particle correlation is studied
intuitively via the two-body densities. Bose-clustering originating from the symmetrization of wave
functions is found, which leads to the appearance of 1-, 2-, and 3-cluster structures. The motion is
divided into being collective and relative, this leads to the establishment of a relation between the
very high vortex states and the low-lying states.
After the experimental realization of the Bose-Einstein
condensation1, various condensates confined under dif-
ferent circumstances have been extensively studied the-
oretically and experimentally. Mostly, the condensates
are considered to be confined in a harmonic trap. Con-
densates trapped by periodic potential have also been
studied due to the appearance of optical lattices.2 It
is believed that the appearance of condensates confined
in particular geometries is possible. Experimentally, the
particle interactions can now be tuned from very weak
to very strong,3−8 it implies that the particle correlation
may become important. Theoretically, to respond, go-
ing beyond the mean field Gross-Pitaevskii (GP) theory
is desirable, and the condensates confined in particular
geometries are also deserved to be considered.
Along this line, in addition to the ground state, the
yrast states have been studied both analytically and
numerically.9−17 The condensation on a ring has also
been studied recently.12 The present paper is also ded-
icated to the N−boson systems confined on a ring with
weak interaction, its scope is broader and covers the
whole low-lying spectra. A similar system has been in-
vestigated analytically by Lieb and Liniger16,17. How-
ever, the emphasis of their papers is different from the
present one, which is placed on analyzing the structures
of the excited states to find out their distinctions and sim-
ilarities, and to find out the modes of excitation. Based
on the analysis, an effort is made to classify the ex-
cited states. Traditionally, the particle correlation and
its effect on the geometry of N−boson systems is a topic
scarcely studied if N is large. In this paper, the corre-
lation is studied intuitively so as the geometric features
inherent in the excited states can be understood. Tradi-
tionally, a separation between the collective and internal
motions is seldom to be considered if N is large. In this
paper such a separation is made and leads to the estab-
lishment of a relation between the vortex states and the
low-lying states.
It is assumed that the N identical bosons confined on
a ring have mass m, spin zero, and square-barrier inter-
action. The ring has a radius R, N is given at 100, 20
and 10000. Let G = ~2/(2mR2) be the unit of energy.
The Hamiltonian then reads
H = −
i<j Vij (1)
where θi is the azimuthal angle of the i-th boson. Vij =
Vo if |θj − θi| ≤ θrange , or = 0 otherwise. Let φk =
eikθ/
2π be a single particle state, −kmax ≤ k ≤ kmax
is assumed. The N−body normalized basis functions
in Fock-representation are |α〉 ≡ |n−kmax , · · · nkmax〉,
where nj is the number of bosons in φj ,
nj = N,
njj = L, the total angular momentum. Then,
H is diagonalized in the space spanned by |α〉, the low-
lying spectrum together with the eigen-wave-functions,
each is a linear combination of |α〉, are thereby obtained.
Let Kα =
2 be the total kinetic energy of an |α〉
state. Evidently, those |α〉 with a large Kα are negligi-
ble for low-lying states. Therefore, one more constraint
Kα ≤ Kmax is further added to control the number of
|α〉. In this procedure, the crucial point is the calcula-
tion of the matrix elements of H . This can be realized
by using the fractional parentage coefficients18 (refer to
eq.(6) below). Numerical results are reported as follows.
This paper concerns only the cases with weak interac-
tion. Firstly, let Vo = 1, θrange = 0.025, and N = 100.
This is corresponding to γ = 0.00157 , where γ is in-
troduced by Lieb and Liniger to measure the strength
of interaction,16,17 this is shown later. When kmax and
Kmax are given at a number of values, the associated
eigen-energiesEj of the first, fifteenth, and sixteenth L =
0 eigen-states are listed in Table I. When (kmax,Kmax)
is changed from (3, 50) to (5,60), the total number of |α〉
is changed from 2167 to 8890. Table I demonstrates that
the great increase of basis functions does not lead to a
remarkable decrease of eigen-energies. Thus the conver-
gency is qualitatively satisfying even for the higher states.
http://arxiv.org/abs/0704.0842v1
TABLE I: Eigen-energies Ej (the unit is G) of the L = 0
states. The first row is (kmax,Kmax), the first column is the
serial number of states j. Vo = 1, θrange = 0.025, and N =
100 are given.
(3,50) (4,50) (4,60) (5,60)
1 39.109 39.090 39.090 39.078
15 53.645 53.616 53.616 53.613
16 54.822 54.800 54.800 54.790
In the following the choice kmax = 4 and Kmax = 50
are adopted, this limitation leads to a 3254-dimensional
space. Thereby the resultant data have at least three
effective figures, this is sufficient for our qualitative pur-
pose.
N=100,  L=0
FIG. 1: The spectrum of L = 0 states, the unit of energy is
G = ~2/(2mR2). N = 100, Vo = 1, and θrange = 0.025 are
assumed, they are the same for Fig.1 to Fig.5. The levels in a
column constitute an oscillation band, the levels in bold line
are doubly degenerate.
The low-lying spectrum is given in Fig.1, where the
lowest fourteen levels are included. Twelve of them can
be ascribed into three bands, in each band the levels
are distributed equidistantly, this is a strong signal of
harmonic-like oscillations. From now on the labels Ψ
and E(L,Z,i) are used to denote the wave function and en-
ergy of the i-th state of the Z-th band (Z=I, II, III,· · ·).
It turns out that the excitation of a pair of particles
both in p-wave but rotating reversely, namely, one par-
ticle in φ1 while the other one in φ−1, is a basic mode,
the pair is called a basic pair in the follows. A number of
such basic pairs might be excited. When 2j particles are
in basic pairs while the remaining N − 2j particles are
in φ0, the associated |α〉 is written as |P (j)〉. For all the
states of the I−band, we found Ψ(0)I,i is mainly a linear
combination of |P (j)〉 together with a small component
denoted by ∆I,i, i.e.,
I,i =
(0,I,i)
(j)〉+∆I,i (2)
where ∆I,i is very small as shown in Table II, while the
coefficients C
(L,Z,i)
j arise from the diagonalization. Thus
the basic structure of the I−band is just a fluctuation of
many of the basic pairs.
TABLE II: The weights of ∆Z,i of the bands with L = 0
i I−band II−band III−band
1 0.009 0.017 0.040
2 0.012 0.030 0.056
3 0.021 0.061 0.088
4 0.035 0.028
5 0.055 0.035
6 0.079 0.106
For lower states, C
(0,Z,i)
j would be very small if j
is larger, e.g., for the ground state, C
(0,I,1)
0 = 0.968
and C
(0,I,1)
j ≥2 ≈ 0, it implies that the excitation of many
pairs is not probable. It also implies that the ground state
wave function obtained via mean-field theory might be a
good approximation. However, for higher states, many
pairs would be excited. E.g., for the third state of the
I−band, C(0,I,3)j = 0.052, 0.406, 0.681,−0.528, and 0.245
when j is from 0 to 4, it implies a stronger fluctuation.
When a |α〉 has not only 2j particles in the basic pairs,
but also m particles in φk, while the remaining particles
in φ0, then it is denoted as |(k)mP (j)〉 (where k = ±1
are allowed) Similarly, we can define |(k1)m1(k2)m2P (j)〉,
and so on. For all the states of the II−band, we found
II,i =
(0,II,i)
[ |(2)1(−1)2P (j)〉
±|(−2)1(1)2P (j)〉] + ∆II,i
where both the + and − signs lead to the same energy,
thus the level is two-fold degenerate. Again, all the ∆II,i
are very small as shown in Table II, thus the fluctua-
tion of basic pairs is again the basic structure. However,
the II−band is characterized by having the additional
3-particle-excitation (one in d-wave and two in p-wave).
For all the states of the third band, we found
III,i
(0,III,i)
j |(2)1(−2)1P (j)〉+∆III,i (4)
Thus, the III−band contains, in addition to the fluc-
tuation of basic pairs, a more energetic pair with each
particle in d-wave. It was found that the spacing
E(0,Z,i+1) − E(0,Z,i) inside all the bands are nearly the
same, they are ∼3.15. This arises because they have the
same mechanism of oscillation, namely, the fluctuation of
basic pairs.
When the energy goes higher, more oscillation bands
can be found. The two extra levels in Fig.1 at the right
are the band-heads of higher bands.
Incidentally, the band-heads of the above three
bands are dominated by |P (0)〉, |(2)1(−1)2P (0)〉 ±
|(−2)1(1)2P (0)〉 and |(2)1(−2)1P (0)〉, respectively, and
their kinetic energies Kα = 0, 6, and 8. Among all the
basis functions with L = 0 and without basic pairs, these
three are the lowest three. This explains why the band-
heads are dominated by them. Once a band-head is fixed,
the corresponding oscillation band would grow up via the
fluctuation of basic pairs.
The particle correlations can be seen intuitively by ob-
serving the two-body densities
ρ2(θ1, θ2) =
dθ3 · · · dθN Ψ(L)∗Z,i Ψ
Similar to the calculation of the matrix elements of
interaction, the above integration can be performed in
coordinate space by extracting the particles 1 and 2
from |α〉 by using the fractional parentage coefficients18,
namely,
|α〉 =
nk(nk − 1)/N(N − 1)φk(1)φk(2)|αk〉
ka,kb
(ka 6=kb)
nkankb/N(N − 1)φka(1)φkb(2)|αkakb〉
where |αk〉 is different from |α〉 by replacing nk with nk−
2, |αkakb〉 is different from |α〉 by replacing nka and nkb
with nka − 1 and nkb − 1, respectively.
0 60 120 180 240 300 360
3 (c)
III-band
3 (b)
II-band
I-band
FIG. 2: ρ2 as functions of θ2 for the I ( a), II ( b), and III (c)
bands of L = 0 states, θ1 = 0 is given. The labels i of the
states Ψ
are marked by the curves.
ρ2 gives the spatial correlation between any pair of
particles as shown in Fig.2. For the ground state Ψ
ρ2 is flat implying that the correlation is weak. However,
it is a little larger when the two particles are opposite to
each other (θ1 = 0 and θ2 = π). It implies the existence
of a weak correlation which is entirely ignored by the
mean field theory. Thus, even the interaction adopted is
weak and even for the ground state, there is still a small
revision to the mean field theory. For higher states of the
I−band, the fluctuation of basic pairs becomes stronger.
Due to the fluctuation, the particles tend to be close to
each other to form a single cluster. This tendency is
clearly shown in Fig.2a.
For the first state of the III − band, Ψ(0)
III,1 has two
peaks in ρ2 implying a 2-cluster structure. It arises from
the two d-wave paticles inherent in the band. The fea-
ture of Ψ
II,1 is lying between Ψ
I,1 and Ψ
III,1. For all
higher states of every band, due to the strong fluctuation
of basic pairs, all the particles tend to be close to each
other as shown in 2b and 2c.
To understand the physics why the particles tend to be
close to each other, let us study the most important basis
state |P (j)〉. By inserting |P (j)〉 into eq.(5) to replace
and by using (6), ρ2 reads
ρ2(θ1, θ2) =
(2π)2N(N−1) [N(N − 1)− j(4N − 6j)
+4j(N − 2j)(1 + cos(θ1 − θ2)) + 4j2 cos2(θ1 − θ2)]
Where there are four terms at the right, the non-
uniformity arises from the third and fourth terms. The
third term causes the particles to be close to each other
to form a single cluster, while the fourth term causes
the two-cluster clustering. When j is small, the fourth
term can be neglected, and the particles tend to form
a single cluster. However, when j ≈ N/2 , the third
term can be neglected, and the particles tend to form
two clusters. It is noted that, if the symmetrization were
dropped, the density contributed by |P (j)〉 would be uni-
form. The appearance of the clustering originates from
the symmetrization of the bosonic wave functions, there-
fore it can be called as bose-clustering.
For L = 1 states, the lowest energy E(1,I,1) is higher
than E(0,I,1) by 1.606, but lower than E(0,I,2). Thus Ψ
is the true first excited state of the system. A number
of oscillation bands exist as well, the wave functions of
the lowest six bands are found as
(1,I,i)
j |(1)1P (j)〉+∆I,i
(1,II,i)
j |(2)1(−1)1P (j)〉+∆II,i
III,i
(1,III,i)
j |(−2)1(1)3P (j)〉+∆III,i
IV,i =
(1,IV,i)
j |(2)1(−2)1(1)1P (j)〉+∆IV,i
V,i =
(1,V,i)
j |(3)1(−1)2P (j)〉+∆V,i
V I,i =
(1,V I,i)
j |(3)1(−2)1P (j)〉+∆V I,i
Where the weights of all the ∆Z,i ≤ 0.1 if i ≤ 4. Thus,
just as the above L = 0 case, all the bands have the
common fluctuation of basic pairs, but each band has
a specific additional few-particle excitation. The ener-
gies of the band-heads from I to V I are 40.70, 45.42,
48.58, 50.14, 52.04, and 53.58 respectively. Further-
more, the spacing ∼3.15 found above is found again for
all these bands due to having the same mechanism of os-
cillation. The I−band is similar to the above I−band
with L = 0 but having an additional single p-wave ex-
citation, the ρ2 of them are one-one similar. Similarly,
the ρ2 of the IV−band is one-one similar to those of
the above III− band with L = 0. The ρ2 of the II
and III−bands are both similar to those of the above
II−band with L = 0. However, the V and V I−bands
are special due to containing the f-wave excitation, the ρ2
of their band-heads exhibit a 3-cluster structure as shown
in Fig.3. When the energy goes even higher, more higher
oscillation bands will appear. For the above six L = 1
bands, their band-heads are dominated by the |α〉 with
Kα = 1, 5, 7, 9, 11, and 13. Obviously, a higher Kα leads
to a higher band.
0 60 120 180 240 300 360
FIG. 3: ρ2 for selected L = 1 states. θ1 = 0, the (Z, i) labels
are marked by the curves.
In general, all the low-lying states can be classified into
oscillation bands. For all the lower bands disregarding
L, it was found that each band-head is dominated by
a basis function containing a specific few-particle exci-
tation but not containing any basic pairs. The energy
order of the bands is determined by the magnitudes of
Kα associated with the dominant basis function |α〉 of
the band-heads. Once a band-head stands, an oscillation
band will grow up from the band-head simply via the fluc-
tuation of basic pairs. For examples, for L = 2 states, the
dominant |α〉 of the band-heads of the four lowest oscil-
lation bands are |(1)2P (0)〉, |(2)1P (0)〉, |(−2)1(1)4P (0)〉,
and |(3)1(−1)1P (0)〉 with Kα = 2, 4, 8, and 10, respec-
tively.
For L = 3 states, the dominant |α〉 of the band-heads of
the three lowest bands are |(1)3P (0)〉, |(2)1(1)1P (0)〉, and
|(3)1P (0)〉, with Kα = 3, 5, and 9, respectively. Since the
p-, d-, and f-wave appear successively, these band-heads
exhibit 1-cluster, 2-cluster, and 3-cluster structures, re-
spectively, as shown in Fig.4.
0 60 120 180 240 300 360
III,1ρ
FIG. 4: ρ2 for the band-heads of L = 3 states, θ1 = 0.
Furthermore, a −L state can be derived from the
corresponding L state simply by changing every k to
−k, i.e., change the components |(k1)m1(k2)m2P (j)〉 to
|(−k1)m1(−k2)m2P (j)〉, and so on. Therefore Ψ(−L)Z,i =
)∗, and E(−L,Z,i) = E(L,Z,i).
0 2 4 6 8
Yrast line
N=100,   Vo=1,  θrange=0.025
FIG. 5: Energies of the yrast states with L = 0 to 10.
Let us study the yrast states Ψ
I,1 , each is the lowest
one for a given L. The energies of them are plotted in
Fig.5, their wave functions are found as
I,1 =
(L,I,1)
j |(1)LP (j)〉+∆LI,1 (9)
where ∆LI,i is very small. When L is small, the fluc-
tuation of basic pairs is small, and the yrast states are
dominated by the j = 0 component |(1)LP (0)〉.When L is
larger, the weight of the |(1)LP (0)〉 component becomes
smaller. E.g., when L = 0, 2, 4, and 10, the weights
of |(1)LP (0)〉 are 0.94, 0.84, 0.75, and 0.54, respectively.
Evidently, the energy going up linearly in the yrast line
in Fig.5 is mainly due to the linear increase of the number
of p-wave particles.
N=20,   L=0
III IV V
FIG. 6: The spectrum of the L = 0 states with N = 20,
Vo = 1, and θrange = 0.025. Refer to Fig.1
When N = 20, all the above qualitative features re-
main unchanged. Examples are given in Fig.6 and 7 to
be compared with Fig.1 and 4. Nonetheless, the decrease
of N implies that the particles have a less chance to meet
each other, thus the particle correlation is expected to be
weaker. Quantitatively, it was found that (i) The spac-
ing of adjacent oscillation levels becomes smaller, it is
now ∼2.2 to replace the previous 3.15 (ii) The fluctu-
ation becomes weaker. E.g., the weights of |P (j)〉 of
the Ψ
I,3 state are 0.01, 0.95, and 0.03 for j = 1, 2, and
3, respectively, while these weights would be 0.16, 0.46,
0 60 120 180 240 300 360
0.20 N=20,  L=3 
III,1
FIG. 7: The same as Fig.4 but with N = 20
and 0.28 if N = 100. (iii) When N becomes small, the
geometric features would become explicit. E.g., for the
3-cluster structure, the difference between the maximum
and minimum of ρ2 is ∼0.007 in Fig.4, but ∼ 0.033 in
Fig.7.
The decrease of Vo or θrange was found to cause an
effect similar to the decrease of N , the spectra would
remain qualitatively unchanged. Quantitatively, when
Vo is changed from 1 to 0.1, the spacing inside a band
is changed from ∼ 3.15 to ∼ 2.14, and the fluctuation
becomes much weaker as expected.
In what follows we study the vortex states. For an
arbitrary Lo ≤ N/2, the spectra of the Ψ(N−Lo)Z,i and
Z,i states are found to be identical
15, except the former
shifts upward as a whole by N − 2Lo, namely,
E(N−Lo,Z,i) = E(Lo,Z,i) +N − 2Lo (10)
Furthermore, their ρ2 are found to be identical.
Let us define an operator
X so that the state
X |α〉 is
related to |α〉 by changing every ki in |α〉 to −ki + 1,
i.e., φki(θ) to φ−ki+1(θ) = e
iθφ−ki(θ). We further found
from the numerical data that
(N−Lo)
Z,i =
holds exactly. In fact,
X causes a reversion of rota-
tion of each particle plus a collective excitation. It does
not cause any change in particle correlation, therefore ρ2
remains exactly unchanged. Thus the L large states,
including the vortex states L = N , can be known from
the L small states.
The underlying physics of this finding is the separabil-
ity of the Hamiltonian (it is emphasized that the sepa-
rability is exact as can be proved by using mathematical
induction). Let θcoll =
θi/N, which describes a col-
lective rotation. Then H = − 1
+Hint, where Hint
describes the relative (internal) motions and does not de-
pend on θcoll. Accordingly, E
(L,Z,i) = L2/N + E
(L,Z,i)
int ,
the former is for collective and the latter is for relative
(internal) motions. The eigen-states can be thereby sep-
arated as Ψ
Z,i =
eiLθcoll ψ
(L,Z,i)
int . The feature of the
internal states ψ
(L,Z,i)
int has been studied in [19]. Where
it was found that, for an arbitrary Lo
(N+Lo,Z,i)
int = ψ
(Lo,Z,i)
With these in mind, eq.(10) and (11) can be derived as
follows.
From the separability
(N−L)
ei(N−L)θcoll ψ
(N−L,Z,i)
int (13)
X acts on a wave function with L, from the defini-
tion of
X, L should be changed to −L and an additional
factor
eiθj = eiN θcoll should be added, thus
Z,i =
ei(N−L)θcoll ψ
(−L,Z,i)
Due to (12), the right hand sides of (13) and (14) are
equal, thereby (11) is proved.
Furthermore, since ψ
(−L,Z,i)
int = (ψ
(L,Z,i)
int )
∗, the internal
energy E
(N−L,Z,i)
int = E
(−L,Z,i)
int = E
(L,Z,i)
int . Therefore,
E(N−L,Z,i) − E(L,Z,i) = (N − L)2/N − L2/N = N − 2L.
This recovers eq.(10), the energy difference arises purely
from the difference in collective rotation.
If the particles are tightly confined on the ring, rapidly
rotating state with a large L = JN − Lo would exist,
where J is an integer. Their spectra would remain the
same but shift upward by J(JN − 2Lo) from the spec-
trum with L = Lo, while Ψ
(JN−Lo)
Z,i =
Z,i , where
XJ changes each φki to φ−ki+J . Thus the rapidly rotat-
ing states have the same internal structure as the corre-
sponding lower states but have a much stronger collective
rotation.
When N increases greatly while Vo or θrange decreases
accordingly, the qualitative behaviors remain unchanged.
E.g., when N = 10000 and Vo = 0.01 (θrange remains
unchanged), the spectrum and the wave functions are
found to be nearly the same as the case N = 100 and
Vo = 1, except that the spectrum has shifted upward
nearly as a whole by 3939. This is again a signal that,
for weak interaction and for the ground states, the mean-
field theory is a good approximation.
It is noted that the confinement by a ring is quite differ-
ent from a 2-dimensional harmonic trap. In the latter,
the energy of a particle in the lowest Landau levels is
proportional to its angular momentum k. However, for
the rings, it is proportional to k2. Consequently, higher
partial waves are seriously suppressed and the p-wave ex-
citation becomes dominant. For a harmonic trap it was
found in [10,11] that d- and f-wave excitations are more
important than the p-wave excitation when L is small.
This situation does not appear in our case.
When the zero-range interaction Vij = gδ(θi − θj) is
adopted, The results are nearly the same with those from
the square-barrier interaction if the parameters are re-
lated as g = 2Voθrange (in this choice both interactions
have the same diagonal matrix elements). For an exam-
ple, a comparison is made in Table III. The high simi-
larity between the two sets of data imply that the above
findings are also valid for zero-range interaction.
TABLE III: Eigen-energies of the four lowest L = 0 states
for a system with N = 100 and with zero-range interaction
Vij = 0.05δ(θi − θj) (the unit of energy is G as before). The
weights of the j = 0 components of these states are also
listed. The corresponding results from square-barrier inter-
action with Vo = 1, and θrange = 0.025 are given in the
parentheses.
(L,Z, i) E(L,Z,i) (C
(L,Z,i)
(0, I, 1) 39.0900 (39.0902) 0.9370 (0.9371)
(0, I, 2) 42.2982 (42.2983) 0.0510 (0.0510)
(0, I, 3) 45.4733 (45.4733) <0.02 (<0.02)
(0, II, 1) 47.0076 (47.0074) 0.8372 (0.8373)
The numerical results from using zero-range interac-
tion can be compared with the exact results from solving
integral equations by Lieb and Liniger [16,17]. The vari-
ables γ and e(γ) introduced in [16] are related to those of
this paper as γ = gπ/N and e(γ) = 4π2E/N3 (the unit
of E is G). However, this paper concerns mainly the case
of weak interaction, say, g ≤ 0.05,or γ ≤ 0.00157 (other-
wise, the procedure of diagonalization would not be valid
due to the cutoff of the space). Nonetheless, even γ is
as large as 0.5 (g = 15.9) the evolution of the ground
state energy with N = 100 against γ obtained via di-
agonalization coincide, in the qualitative sense, with the
exact results quite well . This is shown in Fig.8 to be
compared with Fig.3 of [16], where γ is ranged from 0
to 10. In Fig.8, the constraint γ < e(γ) is recovered.
Furthermore, when γ is small, e(γ) against γ appears as
a straight line.
0.0 0.1 0.2 0.3 0.4 0.5
FIG. 8: e(γ) = 4π2E/N3 against γ = gπ/N . N is given
at 100 and E is the ground state energy calculated from the
diagonalization in the unit G.
In summary, a detailed analysis based on the numerical
data of N−boson systems on a ring with weak interac-
tion has been made. The main result is the discovery of
the basic pairs, which exist extensively in all the excited
states and dominates the low-lying spectra. The fluc-
tuation of basic pairs provides a common mechanism of
oscillation, the low-lying states are thereby classified into
oscillation bands. Each band is characterized by having
its specific additional excitation of a few particles. Since
the mechanism of oscillation is common, the level spac-
ings of different bands are nearly equal in a spectrum.
To divide the motion into being collective and relative
provides a better understanding to the relation between
the higher and lower states. The very high vortex states
with L ≈ N can be understood from the correspond-
ing low-lying states because they have exactly the same
internal states.
The particle correlation has been intuitively studied.
particle densities are found to be in general non-uniform,
bose-clustering originating from the symmetrization of
wave functions is found, which leads to the appearance
of one, two, and three clusters. This phenomenon would
become explicit and might be observed if N is small.
Acknowledgment: The support by NSFC under the
grants 10574163 and 90306016 is appreciated.
REFERENCES
1, D.M. Stamper-Kurn, M.R. Andrews, A.P.
Chikkatur, S. Inouye, H.-J. Miesner, J. Stenger, and W.
Ketterle, Phys. Rev. Lett. 80, 2027 (1998)
2, B.P. Anderson and M.A. Kasevich, Science 281,
1686 (1998)
3, J.L. Roberts, et al, Phys. Rev. Lett. 81, 5159
(1998)
4, J. Stenger, et al, Phys. Rev. Lett. 82, 2422 (1999).
5, S.L. Cornish et al., Phys. Rev. Lett. 85, 1795 (2000)
6, M. Greiner et al., Nature (London) 415, 39 (2002)
7, B. Paredes et al., Nature (London) 429, 277 (2004)
8, G.T. Kinoshita, T. Wenger, and D.S. Weiss, Science
305, 1125 (2004)
9, N.K. Wilkin, J.M.F. Gunn, and R.A. Smith, Phys.
Rev. Lett. 80, 2265 (1998)
10, B. Mottelson, Phys. Rev. Lett. 83, 2695 (1999)
11, G.F. Bertsch and T. Papenbrock, 83, 5412 (1999)
12, K. Sakmann, A.I. Streltsov, O.E. Alon, L.S. Ceder-
baum, Phys. Rev. A 72, 033613 (2005)
13, I. Romanovsky, C. Yannouleas, and U. Landman,
Phys. Rev. Lett. 93, 230405 (2004)
14, I. Romanovsky, C. Yannouleas, L.O. Baksmaty,
and U. Landman, Phys. Rev. Lett. 97, 090401 (2006)
15, Yongle Yu, cond-mat/0609711 v1.
16, E.H. Lieb and W.Liniger, Phys, Rev. 130, 1605
(1963)
17, E.H. Lieb, Phys, Rev. 130, 1616 (1963)
http://arxiv.org/abs/cond-mat/0609711
18, F. Bacher and S. Goudsmit, Phys. Rev., 46, 948
(1934)
19, C.G. Bao, G.M. Huang, and Y.M. Liu, Phys. Rev.
B 72, 195310 (2005)
