Coincidence of the oscillations in the dipole transition and in the persistent current of
narrow quantum rings with two electrons
Y. Z. He and C. G. Bao∗
State Key Laboratory of Optoelectronic Materials and Technologies,
and Department of Physics, Sun Yat-Sen University, Guangzhou, 510275, P.R. China
The fractional Aharonov-Bohm oscillation (FABO) of narrow quantum rings with two electrons
has been studied and has been explained in an analytical way, the evolution of the period and
amplitudes against the magnetic field can be exactly described. Furthermore, the dipole transition
of the ground state was found to have essentially two frequencies, their difference appears as an
oscillation matching the oscillation of the persistent current exactly. A number of equalities relating
the observables and dynamical parameters have been found.
PACS numbers: 73.23.Ra, 78.66.-w
* The corresponding author
Quantum rings containing only a few electrons can
be now fabricated in laboratories1,2. When a magnetic
field B is applied, interesting physical phenomena, e.g.,
Aharonov-Bohm oscillation (ABO) and fractional ABO
(FABO)of the ground state (GS) energy Eo and persis-
tent current Jo, have been observed
2−4,13. In the the-
oretical aspect, a number of calculations based on exact
diagonalization5−8, local-spin-density approximation9,10,
and the diffusion Monte Carlo method11 have been
performed. These calculations can in general repro-
duce the experimental data. For examples, in the
calculation of 4-electron ring6,11, the period of oscilla-
tion Φ0/4 found in experiments was recovered (Φ0 =
hc/eisthefluxquantum).
In addition to the oscillations in Eo and Jo, the oscil-
lation in the optical properties is noticeable.16,17. In this
paper a new kind of oscillation found in the dipole tran-
sition of two-electron (2-e) narrow rings is reported. The
emitted (absorbed) photon of the dipole transition of the
GS was found to have essentially two energies, their dif-
ference is exactly equal to hJo, where h is the Planck’s
constant. In other words the difference of the two photon
energies appears as an oscillation which matches exactly
the oscillation of Jo. This finding is approved by both
numerical calculation and analytical analysis as follows.
The narrow 2-e ring is first considered as one-
dimensional, then the effect of the width of the ring is
further evaluated afterward. The Hamiltonian reads
H = T + V12 +HZeeman (1)
G(−i ∂
+Φ)2, G = ~
2m∗R2
where m∗ the effective mass, θj the azimuthal an-
gle of the j − th electron, Φ = πR2B/Φ0, where B
is a magnetic field perpendicular to the plane of the
ring, V12 the e-e Coulomb interaction, HZeeman =
−SZµΦ the well known Zeeman energy where SZ is
the Z-component of the total spin S, and µ = g
πR2Φ0
where g∗ is the effective g-factor and µB is the Bohr
magneton. The interaction is adjusted as 7 V12 =
e2/(2ε
d2 +R2 sin2((θ1 − θ2)/2))
−1, where ε is the di-
electric constant and the parameter d is introduced to
account for the effect of finite thickness of the ring.
We first perform a numerical calculation so that all
related quantities can be evaluated quantitatively. m∗ =
0.063me, ε = 12.4 (for InGaAs), d = 0.05R , and the
units meV , nm , Tesla and Φ0 are used. Accordingly,
G = 604.8/R2,and µ = 33.53/R2.
A set of basis functions φk1k2 = e
i(k1θ1+k2θ2)/2π is in-
troduced to diagonalize the Hamiltonian, where k1 and
k2 must be integers to assure the periodicity, the sum of
k1 and k2 is just the total orbital angular momentum L.
φk1k2 must be further (anti-)symmetrized when S = 0(1).
When about three thousand basis functions are adopted,
accurate solutions (at least six effective digits) can be ob-
tained. The low-lying spectrum is plotted in Fig.1, where
the oscillation of the GS energy and the transition of the
GS angular momentum Lo can be clearly seen.
Let θC = (θ2 + θ1)/2 , and ϕ = θ2 − θ1. Then
H = Hcoll +Hint (2)
whereHcoll =
G(−i ∂
+2Φ)2+HZeeman andHint =
2G(−i ∂
)2+V12, they are for the collective and internal
motions, respectively. Our numerical results lead to the
following points.
(i) Separability: The separability of one-dimensional
ring is well known5. However, for the convenience of the
following description, it is briefly summarized as follows.
Each eigenenergy E can be exactly divided as a sum of
three terms
E = 1
G(L+ 2Φ)2 + Eint − SZµΦ (3)
where the first term is the kinetic energy of collective
motion, Eint is the internal energy .
Since the basis functions can be rewritten as
http://arxiv.org/abs/0704.0070v1
φk1k2 = e
iLθCei
(k2−k1)ϕ/2π (4)
the spatial part of each eigenstate Ψ is strictly separa-
ble as Ψ = 1√
eiL θCψint where the first part describes
the collective motion, while ψint is a normalized internal
state depending only on ϕ. In particular, both Eint and
ψint do not depend on B (or Φ).
0 2 4 6 8
E (meV)
FIG. 1: Low-lying levels of a 2-e ring against Φ/Φ0 in the
FABO region. When Φ/ is positive, Lo is negative, the num-
bers by the curves are −Lo.
(ii) Classification of ψint: When L is even (odd), (k2−
k1)/2 is an integer (half-integer), thus the period of ϕ as
shown in (4) is 2π (4π). Therefore, the periodicity of the
internal states have two choices. In fact, the difference in
the periodicity is closely related to the dependence of the
domains of the new variables θC and ϕ, this point has
been discussed in detail in ref.[14,15]. Let Q = (−1)L,
then the four cases (Q,S) = (1,0), (-1,0), (-1,1) and (1,1)
are associated with four types of states labeled by a, b, c,
and d , respectively. The internal states of Type a
are denoted as ψa, ψa∗ , · · · and the associated internal
energies as Ea < Ea∗ , · · · and so on. Examples of
ψint and Eint are plotted in Fig.2 and listed in Table 1,
respectively.
Table 1, The lowest and second lowest internal
energies (in meV ) of Type a to d, R = 30nm.
Type a b c d
Eint 2.626 4.247 2.630 4.272
E∗int 6.342 8.912 6.435 9.158
Due to the e-e repulsion, a dumbbell shape (DB), i.e.,
ϕ = 180◦, is advantageous in energy because the two
electrons are farther away from each other meanwhile.
However, a rotation of this geometry by π is equivalent
to an interchange of particles, these operations will cre-
ate the factors (−1)L and (−1)S, respectively, from the
wave function. Therefore, the equivalence leads to a con-
straint, accordingly the DB is allowed only for the states
Type a Type b
0 90 180 270 360
Type c
0 90 180 270 360
Type d
FIG. 2: Four types of ψint against ϕ , R = 40nm. The lowest
three of each type are shown, the higher state has more nodes.
with L + S even, i.e., only for Type a and c. Other-
wise, the states would have an inherent node at the DB
and therefore be higher in energy as shown in Table 1,
where Ea << Eb, Ec << Ed, and Ea ≈ Ec. In Fig.2
the patterns of Type a are one-to-one similar to Type c
, they all have a peak at the DB. On the contrary, all
those of Type (b) and(d) have the inherent node at the
DB. It is noticeable that Type b and c are not continuous
at ϕ = 0 and 2π due to their periods are not equal to 2π.
It was found that the internal states of all the GSs are
either ψa or ψc without exceptions because the favorable
DB is allowed in them. When the dynamical parame-
ters vary in reasonable ranges, the qualitative features of
Fig.2 remain the same.
According to (3), an appropriate Lo would be chosen to
minimize the GS energy. When Φ increases, Lo will un-
dergo even-odd transitions repeatedly and become more
negative as shown in Fig.1. Correspondingly, the total
spin So undergoes singlet-triplet transitions, and ψa and
ψc appear in the GS alternatively. However, due to the
Zeeman effect, when Φ is larger than a critical value Φcrit
, only So = 1 states will be dominant, and accordingly
only ψc will appear in the GS. The region Φ < (>) Φcrit
is called the FABO (ABO) region.
(iii) Persistent Current : Let J1 be the current of the
particle e1. The expression of J1 is well known.
5 However,
since it does not depend on the azimuthal angle, it equals
to its average over θ1. Thus the total current J = J1+J2
J = 1
dθ1dθ2 [Ψ
∗(−i ∂
+2Φ)Ψ+c.c.] (5)
where g = ~/(m∗R2). Using the arguments θC and ϕ
and making use of the separability, the integration over
θC and ϕ can be performed. Thus we have
J = g(L+ 2Φ)/2π (6)
This equation demonstrates explicitly the mechanism
of the oscillation of the persistent current, it is caused
by the step-by-step transition of L during the increase
of Φ. Examples of J are shown in Fig.3, where each
stronger oscillation (associated with a L odd and S = 1
GS) is followed by a weaker oscillation (associated with
a L even and S = 0 GS).
0 2 4 6 8
0 2 4 6 8
0 2 4 6 8
FIG. 3: The oscillation of the persistent current and the two
photon energies of the ground states against Φ/Φ0 in the
FABO region. The unit of current is 10−5C/R, where C
is the velocity of light. In the lowest panel, the black square
(white circle) denotes ~ω+ (~ω−), namely, the energy associ-
ated with Lo to Lo + 1 (Lo − 1) transition.
(iv) Relations among the internal states : Define
Om = e
im(θ1−θC)+ eim(θ2−θC) = 2 cos(mϕ/2). By an-
alyzing the numerical data, we found
Ñ(O1ψa) = ψb + ξa and Ñ(O1ψc) = ψd + ξc (7)
where Ñ is the operator of normalization, both ξa and
ξc are very small functions and depend on the dynamical
parameters very weakly. E.g., when R varies from 30 to
90, the weights of ξa and ξc vary from 0.0004 to 0.0002.
They are so small that in fact can be neglected. Since
O1 contains a node at the DB, it must cause a change of
type from a to b, or from c to d. Thus it is not surprising
that (7) holds. Since O1 is the operator of the dipole
transition (see below), eq.(7) provides an additional rule
of selection as discussed later.
(v) Dipole transition: The probability of dipole tran-
sition reads P
(o),± =
(ω±/c)
3R2 |A
(o),±|
2, where ω±
is the frequency of the photon,
= 〈Ψ(f)±|e
±iθ1 + e±iθ2 |Ψ(o)〉
= δL(f), L(o) ±1〈ψ
int |O1|ψ
int〉 (8)
where (f) and (o) denote the final and initial states,
respectively, the signs ± are associated with L(f) =
L(o) ± 1.
Let the initial state be the GS with Lo, then ψ
must be ψa or ψc depending on Lo is even or odd .
Let α denotes the type of the initial state. Due to (7)
int |O1|ψ
int〉 = δ(f),α < O1ψα|O1ψα >
1/2, where
δ(f),α implies that the final state must be ψb (ψd) if α = a
(c), otherwise the amplitude is zero. Thus, due to the ad-
ditional rule of selection eq.(7), the dipole strength of the
GS is completely concentrated in two final states having
L(f) = L(o) ± 1 and both having the same internal state
specified by eq.(7). Accordingly, only the photons with
the two energies
~ω± = E(f)± − E(o)
= G[ 1
(1± 2(Lo + 2Φ)) + ∆α/G] (9)
can be emitted (absorbed), where ∆α = Eb − Ea or
Ed − Ec depending on α = a or (c). The oscillation
of ~ω± is plotted in the lowest panel of Fig.3. It turns
out that ∆α/G depends on R very weakly, thus ~ω± is
nearly proportional to R−2. Accordingly, a smaller ring
will have a larger probability of transition with a higher
energy.
(vi) FABO region: The oscillation in this region is com-
plicated as shown in Fig.1 and 3. It is noted that the GS
energy (3), persistent current (6), and the photon ener-
gies (9) all contain the factor Lo + 2Φ, thus their FABO
are completely in phase and have the same mechanism
caused by the transition of Lo against Φ. In Fig.1 the
abscissa Φ can be divided into segments, in each the
GS has a specific Lo and the GS energy is given by a
piece of a parabolic curve. The segment is called an
even (odd) segment if Lo is even (odd). At the border of
two neighboring segments the two GS energies are equal.
From the equality and based on (3), the right and left
boundaries of the segment with Lo can be obtained as
Φright(Lo) = (1−(µ/2G)
2)−1[ 1−µ(Ec−Ea)/G
2Lo+(−1)
Lo(2(Ec−Ea)+µ(Lo−1/2))/G]/4 (10)
Φleft(Lo) = (1 − (µ/2G)
2)−1[−1 − µ(Ec − Ea)/G
2Lo − (−1)
Lo(2(Ec − Ea) + µ(Lo + 1/2))/G]/4 (11)
where Lo ≤ 0 and Φright(Lo) = Φleft(Lo − 1), µ arises
from the HZeeman. The length of the segment reads
dLo = Φright(Lo) − Φleft(Lo) = (1 − (µ/2G)
2)−1[1 +
(−1)Lo(2(Ec − Ea) + µLo)/G]/2 (12)
which is related to the period of the FABO. When Φ
increases, the magnitude of Lo would increase. Since µLo
is negative, it is clear from eq.(12) that the length of even
(odd) segments would become shorter (longer) when Φ
increases.
The location of a segment with a given Lo can be
known from the inequality Φleft(Lo) ≤ Φ ≤ Φright(Lo).
Once the relation between Lo and the segments of Φ is
clear, every details of the FABO can be analytically and
exactly explained via the eq.(3), (6), and (9). In particu-
lar, the extrema in each segment can be known by giving
Φ = Φright or Φleft. For an example, the maximal cur-
rent is g(Lo+2Φright)/2π. Incidentally, the minimum of
the GS energy in a segment is Emin = Ec−µ
2/8G+µLo/2
(if So = 1), or just equal to Ea (if S0 = 0).
It is noted that Ec − Ea (cf. Table 1) and µ/G (it
is 0.0554 in our case) are both small. When Φ is small
the magnitude of |Lo| would be also small . In this case
eq.(12) leads to dLo ≈ 1/2, i.e., the period is a half of
the one of the normal ABO. In fact, (12) provides an
quantitative description of the variation of the period of
the FABO.
(vii) ABO region: When Φ becomes sufficiently large,
Lo will become very negative, the even segments will
disappear due to their lengths dLo ≤ 0 . We can de-
fine a critical odd integer Lcrit so that dLcrit−1 ≤ 0
while dLcrit+1 > 0, thereby the critical flux separating
the FABO and ABO region can be defined as
Φcrit = Φleft(Lcrit) (13)
Once Φ > Φcrit, Lo remains odd and the system keeps
polarized. Let IX be the largest even integer smaller
than −(G + 2(Ec − Ea))/µ. It turns out from eq.(12)
that Lcrit = IX + 1. With our parameters, Lcrit = −19
and accordingly Φcrit = 9.003 ( refer to Fig.1). Both
Lcrit and Φcrit depend on R very weakly, but sensitively
on the effective mass m∗.
In the ABO region (Φ > Φcrit), eqs.(10) to (12) do not
hold. Instead we have Φright = −(Lo − 1)/2, Φleft =
−(Lo + 1)/2, and dLo = 1. Thus the normal ABO re-
covers. Evaluated from (6), the magnitude of current is
from −g/2π to g/2π (for a comparison, it is from −g/4π
to g/4π for 1-e rings). From (9) the photon energies ~ω+
is from ∆c −G/2 to ∆c + 3G/2, at the same time ~ω−
is from ∆c + 3G/2 to ∆c −G/2.
(viii) Relations between the photon energies and other
physical quantities : Due to (7), the emitted (absorbed)
dipole photon has only two frequencies , therefore it is
meaningful to define ∆~ω = ~(ω+ − ω−). Directly from
(9) and (6), we have
∆~ω = hJo (14)
where h is the Planck’s constant and Jo is the persis-
tent current of the GS. To compare with 1-e rings, the
latter has ∆~ω = 2hJoĖq.(14) demonstrates that the os-
cillation of ∆~ω and the oscillation of Jo are matched
with each other exactly, they keep strictly proportional
to each other during the variation of Φ.
The maxima of ∆~ω measured in the ABO and FABO
regions, respectively, read
(∆~ω)
max = 2G (15)
(∆~ω)
max = 2G(Lo + 2Φright) (16)
Obviously, (15) provides a way to determine G, m∗can
be thereby obtained. (16) can be rewritten as
Ec−Ea = (G−µLo)/2−(2G−µ)/(4G)(∆~ω)
max (17)
This equation can be used to determine Ec −Ea. Fur-
thermore, we define
Γ~ω = ~(ω+ + ω−) = G+ 2∆α (18)
Once G has been known, (18) can be used to determine
Eb−Ea andEd−Ec. Since the spectrum can be generated
from the internal energies via (3), the evolutions of the
spectrum and the persistent current against Φ can be
understood simply by measuring the photon energies.
0.0 0.2 0.4 0.6 0.8 1.0
-0.15
-0.10
-0.05
B (Tesla)
50 to 120nm
FIG. 4: Evolution of hJo(solid line) and ∆~ω (dotted line)
against B for a 2-e ring with ra = 50 and rb = 120nm
(ix) Effect of the width: We now consider a two-
dimensional model in which the two electrons are strictly
confined in an annular region by a potential U(r), which
is zero if ra < r < rb or is infinite otherwise. Under
this model we have performed numerical calculation to
obtain ∆~ω and hJo, where Jo is now the total angular
current inside the ring (from ra to rb). The result is
shown in Fig.4 where ra = 50 and rb = 120 are assumed,
and the two quantities are slightly different from each
other. However, when the width becomes smaller, say
rb − ra < 30, the two curves overlap. Thus (14) works
not only for one-dimensional but also for two-dimensional
narrow rings. Let us define r = ~/
m∗(∆~ω)ABmax.
For one-dimensional rings and from (15), we have r= R,
where R is the radius of the ring. For two-dimensional
rings, it was found from our numerical calculation that
r ≈ (rb + ra)/2 if rb − ra < 30. E.g., when rb = 100
and ra = 70, r =85.03. When rb = 100 and ra = 90, r
=95.00. Thus (15) works also well for two-dimensional
narrow rings if the R in G is replaced by the average
radius.
It is noted that the band-structure and related optical
properties of 2-e rings have already been studied in de-
tail by Wendler and coauthors18. They classify the eigen-
states according to their radial motion, relative angular
motion, and collective rotation. In our paper the relative
angular motion is further classified into four types ac-
cording to the inherent nodal structures and periodicity
of their wave functions, i.e., according to whether the DB
shape is allowed and whether the wave function is contin-
uous at ϕ = 2π. The DB-accessibility turns out to be im-
portant because it affects the eigenenergies decisively. In
fact, the classification of states based on inherent nodal
structures was found to be crucial in atomic physics,19
this would be also true in two-dimensional systems. Fur-
thermore, the rule of selection for the dipole transition
has been proposed in ref.[18]. In our paper, an addi-
tional rule (namely,eq.(7)) is further proposed based on
the possible transition of internal structures. This rule
would affect the dipole spectrum seriously because the
emission (absorption) is thereby concentrated into two
frequencies. The difference of these two frequencies turns
out to be proportional to the persistent current. There-
fore the measurement of this difference can be used to
determine the magnitude of the current.
In summary, we have studied the FABO both analyt-
ically and numerically. The analytical formalism pro-
vides not only a base for qualitative understanding, but
also provides a number of formulae for quantitative de-
scription. The domain of Φ is divided into segments,
each corresponds to a Lo. This division describes ex-
actly how Lo would transit against |Phi, which causes
directly the FABO. Thereby the variation of the period
and amplitude of the oscillation of the GS energy, persis-
tent current, and the frequencies of dipole transition in
the FABO region can be described exactly. A number of
equalities to relate the physical quantities and dynamical
parameters have been found. In particular, a new oscilla-
tion, namely, the oscillation of ∆~ω was found to match
exactly the oscillation of Jo. Since the photon energies
can be more accurately measured, other observables and
parameters can be thereby determined via the equali-
ties. Since the separability of the Hamiltonian and the
existence of inherent nodes are common, the above de-
scription can be more or less generalized to N−electron
rings, this deserves to be further studied.
Acknowledgment, This work is supported by the
NSFC of China under the grants 10574163 and 90306016.
.REFERENCES
1, A. Lorke, R.J. Luyken, A.O. Govorov, J.P. Kot-
thaus, J.M. Garcia, and P.M. Petroff, Phys. Rev. Lett.
84, 2223 (2000).
2, U.F. Keyser, C. Fühner, S. Borck, R.J. Haug, M.
Bichler, G. Abstreiter, and W. Wegscheider, Phys. Rev.
Lett. 90, 196601 (2003)
3, D. Mailly, C. Chapelier, and A. Benoit, Phys. Rev.
Lett. 70, 2020 (1993)
4, A. Fuhrer, S. Lüscher, T. Ihn, T. Heinzel, K. Ensslin,
W. Wegscheider, and M. Bichler, Nature (London) 413,
822 (2001)
5, S. Viefers, P. Koskinen, P.Singha Deo, M. Manninen,
Physica E 21, 1(2004).
6, K. Niemelä, P. Pietiläinen, P. Hyvönen, and T.
Chakraborty, Europhys. Lett. 36, 533 (1996)
7, M. Korkusinski, P. Hawrylak, and M. Bayer, Phys.
Stat. Sol. B 234, 273 (2002)
8, Z. Barticevic, G. Fuster, and M. Pacheco, Phys.
Rev. B 65, 193307 (2002)
9, M. Ferconi and G.Vignale, Phys. Rev. B 50, 14722
(1994).
10, Li. Serra, M. Barranco, A. Emperador, M. Pi, and
E. Lipparini, Phys. Rev. B 59, 15290 (1999)
11 A. Emperador, F. Pederiva, and E. Lipparini,
Phys. Rev. B 68, 115312 (2003)
12, C.G. Bao, G.M. Huang, Y.M. Liu, Phys. Rev. B
72, 195310 (2005)
13, A.E. Hansen, A. Kristensen, S. Pedersen, C.B.
Sorensen, and P.E. Lindelof, Physica E (Amsterdam)
12,770 (2002).
14, K. Moulopoulos and M. Constantinou, Phys. Rev.
B. 70, 235327 (2004)
15,J. Planelles, J.I. Climente, and J.L. Movilla,
arXiv:cond-mat/0506691 (2005)
16, J.I. Climente and J. Planelles, Phys. Rev. B 72,
155322 (2005)
17, A.O. Govorov, S.E. Ulloa, K. Karrai, and R.J. War-
burton, Phys. Rev. B 66, 081309 (2002)
18, L. Wendler, V.M. Fomin, A.V. Chaplik, and A.O.
Govorov, Phys. Rev. B 54, 4794 (1996).
19, M.D. Poulsen and L.B. Madsen, Phys. Rev. A 72,
042501 (2005).
http://arxiv.org/abs/cond-mat/0506691

  The fractional Aharonov-Bohm oscillation (FABO) of narrow quantum rings with
two electrons has been studied and has been explained in an analytical way, the
evolution of the period and amplitudes against the magnetic field can be
exactly described. Furthermore, the dipole transition of the ground state was
found to have essentially two frequencies, their difference appears as an
oscillation matching the oscillation of the persistent current exactly. A
number of equalities relating the observables and dynamical parameters have
been found.

Coincidence of the oscillations in the dipole transition and in the persistent current of
narrow quantum rings with two electrons
Y. Z. He and C. G. Bao∗
State Key Laboratory of Optoelectronic Materials and Technologies,
and Department of Physics, Sun Yat-Sen University, Guangzhou, 510275, P.R. China
The fractional Aharonov-Bohm oscillation (FABO) of narrow quantum rings with two electrons
has been studied and has been explained in an analytical way, the evolution of the period and
amplitudes against the magnetic field can be exactly described. Furthermore, the dipole transition
of the ground state was found to have essentially two frequencies, their difference appears as an
oscillation matching the oscillation of the persistent current exactly. A number of equalities relating
the observables and dynamical parameters have been found.
PACS numbers: 73.23.Ra, 78.66.-w
* The corresponding author
Quantum rings containing only a few electrons can
be now fabricated in laboratories1,2. When a magnetic
field B is applied, interesting physical phenomena, e.g.,
Aharonov-Bohm oscillation (ABO) and fractional ABO
(FABO)of the ground state (GS) energy Eo and persis-
tent current Jo, have been observed
2−4,13. In the the-
oretical aspect, a number of calculations based on exact
diagonalization5−8, local-spin-density approximation9,10,
and the diffusion Monte Carlo method11 have been
performed. These calculations can in general repro-
duce the experimental data. For examples, in the
calculation of 4-electron ring6,11, the period of oscilla-
tion Φ0/4 found in experiments was recovered (Φ0 =
hc/eisthefluxquantum).
In addition to the oscillations in Eo and Jo, the oscil-
lation in the optical properties is noticeable.16,17. In this
paper a new kind of oscillation found in the dipole tran-
sition of two-electron (2-e) narrow rings is reported. The
emitted (absorbed) photon of the dipole transition of the
GS was found to have essentially two energies, their dif-
ference is exactly equal to hJo, where h is the Planck’s
constant. In other words the difference of the two photon
energies appears as an oscillation which matches exactly
the oscillation of Jo. This finding is approved by both
numerical calculation and analytical analysis as follows.
The narrow 2-e ring is first considered as one-
dimensional, then the effect of the width of the ring is
further evaluated afterward. The Hamiltonian reads
H = T + V12 +HZeeman (1)
G(−i ∂
+Φ)2, G = ~
2m∗R2
where m∗ the effective mass, θj the azimuthal an-
gle of the j − th electron, Φ = πR2B/Φ0, where B
is a magnetic field perpendicular to the plane of the
ring, V12 the e-e Coulomb interaction, HZeeman =
−SZµΦ the well known Zeeman energy where SZ is
the Z-component of the total spin S, and µ = g
πR2Φ0
where g∗ is the effective g-factor and µB is the Bohr
magneton. The interaction is adjusted as 7 V12 =
e2/(2ε
d2 +R2 sin2((θ1 − θ2)/2))
−1, where ε is the di-
electric constant and the parameter d is introduced to
account for the effect of finite thickness of the ring.
We first perform a numerical calculation so that all
related quantities can be evaluated quantitatively. m∗ =
0.063me, ε = 12.4 (for InGaAs), d = 0.05R , and the
units meV , nm , Tesla and Φ0 are used. Accordingly,
G = 604.8/R2,and µ = 33.53/R2.
A set of basis functions φk1k2 = e
i(k1θ1+k2θ2)/2π is in-
troduced to diagonalize the Hamiltonian, where k1 and
k2 must be integers to assure the periodicity, the sum of
k1 and k2 is just the total orbital angular momentum L.
φk1k2 must be further (anti-)symmetrized when S = 0(1).
When about three thousand basis functions are adopted,
accurate solutions (at least six effective digits) can be ob-
tained. The low-lying spectrum is plotted in Fig.1, where
the oscillation of the GS energy and the transition of the
GS angular momentum Lo can be clearly seen.
Let θC = (θ2 + θ1)/2 , and ϕ = θ2 − θ1. Then
H = Hcoll +Hint (2)
whereHcoll =
G(−i ∂
+2Φ)2+HZeeman andHint =
2G(−i ∂
)2+V12, they are for the collective and internal
motions, respectively. Our numerical results lead to the
following points.
(i) Separability: The separability of one-dimensional
ring is well known5. However, for the convenience of the
following description, it is briefly summarized as follows.
Each eigenenergy E can be exactly divided as a sum of
three terms
E = 1
G(L+ 2Φ)2 + Eint − SZµΦ (3)
where the first term is the kinetic energy of collective
motion, Eint is the internal energy .
Since the basis functions can be rewritten as
http://arxiv.org/abs/0704.0070v1
φk1k2 = e
iLθCei
(k2−k1)ϕ/2π (4)
the spatial part of each eigenstate Ψ is strictly separa-
ble as Ψ = 1√
eiL θCψint where the first part describes
the collective motion, while ψint is a normalized internal
state depending only on ϕ. In particular, both Eint and
ψint do not depend on B (or Φ).
0 2 4 6 8
E (meV)
FIG. 1: Low-lying levels of a 2-e ring against Φ/Φ0 in the
FABO region. When Φ/ is positive, Lo is negative, the num-
bers by the curves are −Lo.
(ii) Classification of ψint: When L is even (odd), (k2−
k1)/2 is an integer (half-integer), thus the period of ϕ as
shown in (4) is 2π (4π). Therefore, the periodicity of the
internal states have two choices. In fact, the difference in
the periodicity is closely related to the dependence of the
domains of the new variables θC and ϕ, this point has
been discussed in detail in ref.[14,15]. Let Q = (−1)L,
then the four cases (Q,S) = (1,0), (-1,0), (-1,1) and (1,1)
are associated with four types of states labeled by a, b, c,
and d , respectively. The internal states of Type a
are denoted as ψa, ψa∗ , · · · and the associated internal
energies as Ea < Ea∗ , · · · and so on. Examples of
ψint and Eint are plotted in Fig.2 and listed in Table 1,
respectively.
Table 1, The lowest and second lowest internal
energies (in meV ) of Type a to d, R = 30nm.
Type a b c d
Eint 2.626 4.247 2.630 4.272
E∗int 6.342 8.912 6.435 9.158
Due to the e-e repulsion, a dumbbell shape (DB), i.e.,
ϕ = 180◦, is advantageous in energy because the two
electrons are farther away from each other meanwhile.
However, a rotation of this geometry by π is equivalent
to an interchange of particles, these operations will cre-
ate the factors (−1)L and (−1)S, respectively, from the
wave function. Therefore, the equivalence leads to a con-
straint, accordingly the DB is allowed only for the states
Type a Type b
0 90 180 270 360
Type c
0 90 180 270 360
Type d
FIG. 2: Four types of ψint against ϕ , R = 40nm. The lowest
three of each type are shown, the higher state has more nodes.
with L + S even, i.e., only for Type a and c. Other-
wise, the states would have an inherent node at the DB
and therefore be higher in energy as shown in Table 1,
where Ea << Eb, Ec << Ed, and Ea ≈ Ec. In Fig.2
the patterns of Type a are one-to-one similar to Type c
, they all have a peak at the DB. On the contrary, all
those of Type (b) and(d) have the inherent node at the
DB. It is noticeable that Type b and c are not continuous
at ϕ = 0 and 2π due to their periods are not equal to 2π.
It was found that the internal states of all the GSs are
either ψa or ψc without exceptions because the favorable
DB is allowed in them. When the dynamical parame-
ters vary in reasonable ranges, the qualitative features of
Fig.2 remain the same.
According to (3), an appropriate Lo would be chosen to
minimize the GS energy. When Φ increases, Lo will un-
dergo even-odd transitions repeatedly and become more
negative as shown in Fig.1. Correspondingly, the total
spin So undergoes singlet-triplet transitions, and ψa and
ψc appear in the GS alternatively. However, due to the
Zeeman effect, when Φ is larger than a critical value Φcrit
, only So = 1 states will be dominant, and accordingly
only ψc will appear in the GS. The region Φ < (>) Φcrit
is called the FABO (ABO) region.
(iii) Persistent Current : Let J1 be the current of the
particle e1. The expression of J1 is well known.
5 However,
since it does not depend on the azimuthal angle, it equals
to its average over θ1. Thus the total current J = J1+J2
J = 1
dθ1dθ2 [Ψ
∗(−i ∂
+2Φ)Ψ+c.c.] (5)
where g = ~/(m∗R2). Using the arguments θC and ϕ
and making use of the separability, the integration over
θC and ϕ can be performed. Thus we have
J = g(L+ 2Φ)/2π (6)
This equation demonstrates explicitly the mechanism
of the oscillation of the persistent current, it is caused
by the step-by-step transition of L during the increase
of Φ. Examples of J are shown in Fig.3, where each
stronger oscillation (associated with a L odd and S = 1
GS) is followed by a weaker oscillation (associated with
a L even and S = 0 GS).
0 2 4 6 8
0 2 4 6 8
0 2 4 6 8
FIG. 3: The oscillation of the persistent current and the two
photon energies of the ground states against Φ/Φ0 in the
FABO region. The unit of current is 10−5C/R, where C
is the velocity of light. In the lowest panel, the black square
(white circle) denotes ~ω+ (~ω−), namely, the energy associ-
ated with Lo to Lo + 1 (Lo − 1) transition.
(iv) Relations among the internal states : Define
Om = e
im(θ1−θC)+ eim(θ2−θC) = 2 cos(mϕ/2). By an-
alyzing the numerical data, we found
Ñ(O1ψa) = ψb + ξa and Ñ(O1ψc) = ψd + ξc (7)
where Ñ is the operator of normalization, both ξa and
ξc are very small functions and depend on the dynamical
parameters very weakly. E.g., when R varies from 30 to
90, the weights of ξa and ξc vary from 0.0004 to 0.0002.
They are so small that in fact can be neglected. Since
O1 contains a node at the DB, it must cause a change of
type from a to b, or from c to d. Thus it is not surprising
that (7) holds. Since O1 is the operator of the dipole
transition (see below), eq.(7) provides an additional rule
of selection as discussed later.
(v) Dipole transition: The probability of dipole tran-
sition reads P
(o),± =
(ω±/c)
3R2 |A
(o),±|
2, where ω±
is the frequency of the photon,
= 〈Ψ(f)±|e
±iθ1 + e±iθ2 |Ψ(o)〉
= δL(f), L(o) ±1〈ψ
int |O1|ψ
int〉 (8)
where (f) and (o) denote the final and initial states,
respectively, the signs ± are associated with L(f) =
L(o) ± 1.
Let the initial state be the GS with Lo, then ψ
must be ψa or ψc depending on Lo is even or odd .
Let α denotes the type of the initial state. Due to (7)
int |O1|ψ
int〉 = δ(f),α < O1ψα|O1ψα >
1/2, where
δ(f),α implies that the final state must be ψb (ψd) if α = a
(c), otherwise the amplitude is zero. Thus, due to the ad-
ditional rule of selection eq.(7), the dipole strength of the
GS is completely concentrated in two final states having
L(f) = L(o) ± 1 and both having the same internal state
specified by eq.(7). Accordingly, only the photons with
the two energies
~ω± = E(f)± − E(o)
= G[ 1
(1± 2(Lo + 2Φ)) + ∆α/G] (9)
can be emitted (absorbed), where ∆α = Eb − Ea or
Ed − Ec depending on α = a or (c). The oscillation
of ~ω± is plotted in the lowest panel of Fig.3. It turns
out that ∆α/G depends on R very weakly, thus ~ω± is
nearly proportional to R−2. Accordingly, a smaller ring
will have a larger probability of transition with a higher
energy.
(vi) FABO region: The oscillation in this region is com-
plicated as shown in Fig.1 and 3. It is noted that the GS
energy (3), persistent current (6), and the photon ener-
gies (9) all contain the factor Lo + 2Φ, thus their FABO
are completely in phase and have the same mechanism
caused by the transition of Lo against Φ. In Fig.1 the
abscissa Φ can be divided into segments, in each the
GS has a specific Lo and the GS energy is given by a
piece of a parabolic curve. The segment is called an
even (odd) segment if Lo is even (odd). At the border of
two neighboring segments the two GS energies are equal.
From the equality and based on (3), the right and left
boundaries of the segment with Lo can be obtained as
Φright(Lo) = (1−(µ/2G)
2)−1[ 1−µ(Ec−Ea)/G
2Lo+(−1)
Lo(2(Ec−Ea)+µ(Lo−1/2))/G]/4 (10)
Φleft(Lo) = (1 − (µ/2G)
2)−1[−1 − µ(Ec − Ea)/G
2Lo − (−1)
Lo(2(Ec − Ea) + µ(Lo + 1/2))/G]/4 (11)
where Lo ≤ 0 and Φright(Lo) = Φleft(Lo − 1), µ arises
from the HZeeman. The length of the segment reads
dLo = Φright(Lo) − Φleft(Lo) = (1 − (µ/2G)
2)−1[1 +
(−1)Lo(2(Ec − Ea) + µLo)/G]/2 (12)
which is related to the period of the FABO. When Φ
increases, the magnitude of Lo would increase. Since µLo
is negative, it is clear from eq.(12) that the length of even
(odd) segments would become shorter (longer) when Φ
increases.
The location of a segment with a given Lo can be
known from the inequality Φleft(Lo) ≤ Φ ≤ Φright(Lo).
Once the relation between Lo and the segments of Φ is
clear, every details of the FABO can be analytically and
exactly explained via the eq.(3), (6), and (9). In particu-
lar, the extrema in each segment can be known by giving
Φ = Φright or Φleft. For an example, the maximal cur-
rent is g(Lo+2Φright)/2π. Incidentally, the minimum of
the GS energy in a segment is Emin = Ec−µ
2/8G+µLo/2
(if So = 1), or just equal to Ea (if S0 = 0).
It is noted that Ec − Ea (cf. Table 1) and µ/G (it
is 0.0554 in our case) are both small. When Φ is small
the magnitude of |Lo| would be also small . In this case
eq.(12) leads to dLo ≈ 1/2, i.e., the period is a half of
the one of the normal ABO. In fact, (12) provides an
quantitative description of the variation of the period of
the FABO.
(vii) ABO region: When Φ becomes sufficiently large,
Lo will become very negative, the even segments will
disappear due to their lengths dLo ≤ 0 . We can de-
fine a critical odd integer Lcrit so that dLcrit−1 ≤ 0
while dLcrit+1 > 0, thereby the critical flux separating
the FABO and ABO region can be defined as
Φcrit = Φleft(Lcrit) (13)
Once Φ > Φcrit, Lo remains odd and the system keeps
polarized. Let IX be the largest even integer smaller
than −(G + 2(Ec − Ea))/µ. It turns out from eq.(12)
that Lcrit = IX + 1. With our parameters, Lcrit = −19
and accordingly Φcrit = 9.003 ( refer to Fig.1). Both
Lcrit and Φcrit depend on R very weakly, but sensitively
on the effective mass m∗.
In the ABO region (Φ > Φcrit), eqs.(10) to (12) do not
hold. Instead we have Φright = −(Lo − 1)/2, Φleft =
−(Lo + 1)/2, and dLo = 1. Thus the normal ABO re-
covers. Evaluated from (6), the magnitude of current is
from −g/2π to g/2π (for a comparison, it is from −g/4π
to g/4π for 1-e rings). From (9) the photon energies ~ω+
is from ∆c −G/2 to ∆c + 3G/2, at the same time ~ω−
is from ∆c + 3G/2 to ∆c −G/2.
(viii) Relations between the photon energies and other
physical quantities : Due to (7), the emitted (absorbed)
dipole photon has only two frequencies , therefore it is
meaningful to define ∆~ω = ~(ω+ − ω−). Directly from
(9) and (6), we have
∆~ω = hJo (14)
where h is the Planck’s constant and Jo is the persis-
tent current of the GS. To compare with 1-e rings, the
latter has ∆~ω = 2hJoĖq.(14) demonstrates that the os-
cillation of ∆~ω and the oscillation of Jo are matched
with each other exactly, they keep strictly proportional
to each other during the variation of Φ.
The maxima of ∆~ω measured in the ABO and FABO
regions, respectively, read
(∆~ω)
max = 2G (15)
(∆~ω)
max = 2G(Lo + 2Φright) (16)
Obviously, (15) provides a way to determine G, m∗can
be thereby obtained. (16) can be rewritten as
Ec−Ea = (G−µLo)/2−(2G−µ)/(4G)(∆~ω)
max (17)
This equation can be used to determine Ec −Ea. Fur-
thermore, we define
Γ~ω = ~(ω+ + ω−) = G+ 2∆α (18)
Once G has been known, (18) can be used to determine
Eb−Ea andEd−Ec. Since the spectrum can be generated
from the internal energies via (3), the evolutions of the
spectrum and the persistent current against Φ can be
understood simply by measuring the photon energies.
0.0 0.2 0.4 0.6 0.8 1.0
-0.15
-0.10
-0.05
B (Tesla)
50 to 120nm
FIG. 4: Evolution of hJo(solid line) and ∆~ω (dotted line)
against B for a 2-e ring with ra = 50 and rb = 120nm
(ix) Effect of the width: We now consider a two-
dimensional model in which the two electrons are strictly
confined in an annular region by a potential U(r), which
is zero if ra < r < rb or is infinite otherwise. Under
this model we have performed numerical calculation to
obtain ∆~ω and hJo, where Jo is now the total angular
current inside the ring (from ra to rb). The result is
shown in Fig.4 where ra = 50 and rb = 120 are assumed,
and the two quantities are slightly different from each
other. However, when the width becomes smaller, say
rb − ra < 30, the two curves overlap. Thus (14) works
not only for one-dimensional but also for two-dimensional
narrow rings. Let us define r = ~/
m∗(∆~ω)ABmax.
For one-dimensional rings and from (15), we have r= R,
where R is the radius of the ring. For two-dimensional
rings, it was found from our numerical calculation that
r ≈ (rb + ra)/2 if rb − ra < 30. E.g., when rb = 100
and ra = 70, r =85.03. When rb = 100 and ra = 90, r
=95.00. Thus (15) works also well for two-dimensional
narrow rings if the R in G is replaced by the average
radius.
It is noted that the band-structure and related optical
properties of 2-e rings have already been studied in de-
tail by Wendler and coauthors18. They classify the eigen-
states according to their radial motion, relative angular
motion, and collective rotation. In our paper the relative
angular motion is further classified into four types ac-
cording to the inherent nodal structures and periodicity
of their wave functions, i.e., according to whether the DB
shape is allowed and whether the wave function is contin-
uous at ϕ = 2π. The DB-accessibility turns out to be im-
portant because it affects the eigenenergies decisively. In
fact, the classification of states based on inherent nodal
structures was found to be crucial in atomic physics,19
this would be also true in two-dimensional systems. Fur-
thermore, the rule of selection for the dipole transition
has been proposed in ref.[18]. In our paper, an addi-
tional rule (namely,eq.(7)) is further proposed based on
the possible transition of internal structures. This rule
would affect the dipole spectrum seriously because the
emission (absorption) is thereby concentrated into two
frequencies. The difference of these two frequencies turns
out to be proportional to the persistent current. There-
fore the measurement of this difference can be used to
determine the magnitude of the current.
In summary, we have studied the FABO both analyt-
ically and numerically. The analytical formalism pro-
vides not only a base for qualitative understanding, but
also provides a number of formulae for quantitative de-
scription. The domain of Φ is divided into segments,
each corresponds to a Lo. This division describes ex-
actly how Lo would transit against |Phi, which causes
directly the FABO. Thereby the variation of the period
and amplitude of the oscillation of the GS energy, persis-
tent current, and the frequencies of dipole transition in
the FABO region can be described exactly. A number of
equalities to relate the physical quantities and dynamical
parameters have been found. In particular, a new oscilla-
tion, namely, the oscillation of ∆~ω was found to match
exactly the oscillation of Jo. Since the photon energies
can be more accurately measured, other observables and
parameters can be thereby determined via the equali-
ties. Since the separability of the Hamiltonian and the
existence of inherent nodes are common, the above de-
scription can be more or less generalized to N−electron
rings, this deserves to be further studied.
Acknowledgment, This work is supported by the
NSFC of China under the grants 10574163 and 90306016.
.REFERENCES
1, A. Lorke, R.J. Luyken, A.O. Govorov, J.P. Kot-
thaus, J.M. Garcia, and P.M. Petroff, Phys. Rev. Lett.
84, 2223 (2000).
2, U.F. Keyser, C. Fühner, S. Borck, R.J. Haug, M.
Bichler, G. Abstreiter, and W. Wegscheider, Phys. Rev.
Lett. 90, 196601 (2003)
3, D. Mailly, C. Chapelier, and A. Benoit, Phys. Rev.
Lett. 70, 2020 (1993)
4, A. Fuhrer, S. Lüscher, T. Ihn, T. Heinzel, K. Ensslin,
W. Wegscheider, and M. Bichler, Nature (London) 413,
822 (2001)
5, S. Viefers, P. Koskinen, P.Singha Deo, M. Manninen,
Physica E 21, 1(2004).
6, K. Niemelä, P. Pietiläinen, P. Hyvönen, and T.
Chakraborty, Europhys. Lett. 36, 533 (1996)
7, M. Korkusinski, P. Hawrylak, and M. Bayer, Phys.
Stat. Sol. B 234, 273 (2002)
8, Z. Barticevic, G. Fuster, and M. Pacheco, Phys.
Rev. B 65, 193307 (2002)
9, M. Ferconi and G.Vignale, Phys. Rev. B 50, 14722
(1994).
10, Li. Serra, M. Barranco, A. Emperador, M. Pi, and
E. Lipparini, Phys. Rev. B 59, 15290 (1999)
11 A. Emperador, F. Pederiva, and E. Lipparini,
Phys. Rev. B 68, 115312 (2003)
12, C.G. Bao, G.M. Huang, Y.M. Liu, Phys. Rev. B
72, 195310 (2005)
13, A.E. Hansen, A. Kristensen, S. Pedersen, C.B.
Sorensen, and P.E. Lindelof, Physica E (Amsterdam)
12,770 (2002).
14, K. Moulopoulos and M. Constantinou, Phys. Rev.
B. 70, 235327 (2004)
15,J. Planelles, J.I. Climente, and J.L. Movilla,
arXiv:cond-mat/0506691 (2005)
16, J.I. Climente and J. Planelles, Phys. Rev. B 72,
155322 (2005)
17, A.O. Govorov, S.E. Ulloa, K. Karrai, and R.J. War-
burton, Phys. Rev. B 66, 081309 (2002)
18, L. Wendler, V.M. Fomin, A.V. Chaplik, and A.O.
Govorov, Phys. Rev. B 54, 4794 (1996).
19, M.D. Poulsen and L.B. Madsen, Phys. Rev. A 72,
042501 (2005).
http://arxiv.org/abs/cond-mat/0506691
