arXiv:0809.3532v3 [cond-mat.supr-con] 7 Jun 2010
Spontaneous generation of a half-quantum vortex in spin-orbit coupled Bose-Einstein
Congjun Wu1and Ian Mondragon-Shem1,2
1Department of Physics, University of California, San Diego, CA 92093
2Instituto de F´ ısica, Universidad de Antioquia, AA 1226, Medell´ ın, Colombia
According to the “no-node” theorem, many-body ground state wavefunctions of conventional
Bose-Einstein condensation (BEC) are positive-definite, thus time-reversal symmetry cannot be
spontaneously broken. We find that multi-component bosons with spin-orbit coupling provide an
unconventional type of BEC beyond this paradigm. In a harmonic confining trap, the condensate
spontaneously generates a half-quantum vortex combined with the skyrmion type of spin texture.
This phenomenon can be realized for both the cold atom systems with artificial spin-orbit coupling
generated from atom-laser interaction, and for exciton condensates in semi-conductor systems.
PACS numbers: 71.35.-y, 73.50.-h, 03.75.Mn, 03.75.Nt
The conventional many-body ground state wavefunc-
tions of bosons satisfy the celebrated “no-node” theorem
in the absence of rotation, as pointed out by Feynman
, which means that they are positive-definite in the
time-reversal (TR) symmetry cannot be spontaneously
broken.It applies to various quantum ground states
of bosons including Bose-Einstein condensates (BEC),
Mott-insulating states, density-wave states, and super-
solid states, thus making this a very general statement.
It would be interesting to search for novel types of
quantum states of bosons beyond this paradigm, such
as unconventional BECs with complex-valued wavefunc-
tions. The “no-node” theorem is only a ground state
property, hence it does not apply to excited states. Re-
cently, a meta-stable state of bosons has been realized,
using cold atoms in optical lattices, by pumping atoms
into high orbital bands (e.g. p-bands) with a long life
time [2, 3]. Interactions among orbital bosons are char-
acterized by an orbital Hund’s rule [4, 5], which gives
rise to a class of complex superfluid states by develop-
ing onsite orbital angular momentum, thus breaking TR
symmetry spontaneously [3–7].
The “no-node” theorem does not apply to spinful
bosons with spin-orbit (SO) coupling either. The linear
dependence on momentum introduced by the SO cou-
pling invalidates Feynman’s proof. Although alkali atoms
are too heavy to exhibit relativistic SO coupling, artificial
SO coupling can be induced by laser-atom interactions
[8–13]. The BECs in such systems have been studied,
and have been shown to exhibit frustrations of superflu-
idity . On the other hand, excitons in semiconductors
[15–18] with small effective mass can exhibit SO coupling
in the center-of-mass motion [19–21]. In particular, ex-
citing progress has been made in indirect exciton systems
in coupled quantum wells where electrons and holes are
spatially separate [22–25]. The extraordinarily long life
time of indirect excitons provides a wonderful opportu-
nity to investigate the exotic state of matter of exciton
This theorem implies that
In this article, we show that bosons with Rashba SO
coupling provide a new opportunity to investigate un-
conventional BEC beyond the “no-node” theorem. We
find that Rashba SO coupling leads to fragmented con-
densation in homogeneous systems. Furthermore, with
an external harmonic potential trap, TR symmetry is
spontaneously broken. In this case, the ground state
condensate develops orbital angular momentum, which
can be viewed as a half-quantum vortex. Moreover, the
spin density distribution exhibits a cylindrically symmet-
ric spiral pattern as skyrmions. The experimental real-
ization, in both cold atom and semiconductor exciton
systems, is discussed.
We begin with a 3D two-component boson system with
Rashba SO coupling in the xy-plane and with contact
interaction, described by
d3? r ψ†
− µ]ψα+ ?λRψ†
where ψα is the boson operator; the pseudospin indices
α =↑,↓ refer to two different internal components of
the bosons; λRis the SO coupling strength; g describes
the s-wave scattering interaction. Although Eq. (1) is
for bosons, it satisfies a suitably defined TR symmetry
T = iσ2C satisfying T2= −1, where C is the complex
conjugate operation and σ2operates on the boson pseu-
dospin degree of freedom.
In the homogeneous system, the single particle states
are the helicity eigenstates of ? σ · (?k × ˆ z) which have a
dispersion relation given by ǫ±(?k) =
where k0 =
. The energy minima are located on
the lower branch along a ring with radius k0 in the
plane of kz = 0.The corresponding two-component
wavefunction ψ+(?k) with |k| = k0 can be solved as
angle of the projection of?k in the xy-plane. The in-
teraction part in Eq. (1) in the helicity basis can be
√2(e−iφk/2,ieiφk/2), where φk is the azimuth
FIG. 1: The degeneracy of the single particle ground states
along a circle with the radius k0 in momentum space. We
assume that the fragmented condensation occurs at points A
(k0ˆ ex,0,0) and B (−k0ˆ ex,0,0) with spin polarizations along
±ˆ ey, respectively. The low energy excitations within |k−k0| <
Λ, |kz| < Λ, and Λ/k0 ≪ 1 are classified into two regimes I
(inside two cylinders centering around A and B with radius
of Λ) and II (outside).
?? p1+ ? q;λ|? p1;ρ??? p2− ? q;µ|? p2;ν?
λ(? p1+ ? q)ψ†
µ(? p2− ? q)ψν(? p2)ψρ(? p1), (2)
where the Greek indices λ,ν,µ,ρ are the helicity indices
±; the matrix elements denote the inner product of spin
wavefunctions of two helicity eigenstates at different mo-
menta, e.g., ?? p1+ ? q;λ|? p1;ρ? =1
At the Hartree-Fock level, the kinetic energy can be
minimized through coherent or fragmented condensation
in the linear superpositions of the single particle ground
states.For the interaction energy, the exchange part
among bosons is positive within the s-wave scattering ap-
proximation where g is independent of momentum, thus
the ground state favors the zero exchange energy. The
spin polarizations at?k and −?k in the plane of kz= 0 are
orthogonal to each other. Thus, without loss of gener-
ality, the fragmented condensate defined below satisfies
2[1 + λρei(φp1−φp1+q)].
where?kA= (−k0,0,0) and?kB= (k0,0,0); and (NA,NB)
is the particle number partition satisfying NA+NB= N0,
with N0the total particle number in the condensate. In
homogeneous systems, fragmented condensates with dif-
ferent partitions do not mix because of momentum con-
An “order from disorder” calculation on the low energy
excitation spectra is needed to determine the optimal
particle number partition. The momentum cutoff ?Λ is
defined as ?2Λ2/(2M) ≈ gn, where gn is the interaction
energy scale and n is the particle density. We only con-
sider the limit of strong SO coupling, k0≫ Λ, and leave
the general case for later study. The low energy Bogoli-
ubov excitations that significantly mix particle and hole
operators mainly lie in the regime |
and |kz| < Λ, which can be further divided into two parts
I and II as depicted in Fig.1. Part I is inside two cylin-
ders with the radius of Λ centering around A and B, and
part II is outside these two cylinders.
For the low energy states in part I, we define
the bosonic operators in the lower branch as a? q =
ψ+(−k0ˆ ex+? q) and b? q= ψ+(k0ˆ ex+? q), and ?aq=0? =√Na
and ?bq=0? =√Nb respectively. The relative phase be-
tween two condensates can be set to zero due to a suit-
able translation operation. The low energy excitations in
this region have been calculated in Ref. . By defin-
1(? q) =
the quadratic level of q, can be represented as
y− k0| < Λ
2(? q) =
q), the mean-field Hamiltonian in region I, up to
1(? q)γ(? q) + γ†
2(? q)γ2(? q)?
1(−? q) + h.c.?,
where E(±? q) = ?(q2
ter the Bogoliubov diagonalization yields the spectra
?E? q(E? q+ 2gN0) ≈
the phonon mode describing the overall density fluctua-
tions, which exhibits linear dispersion relation for ? q in the
xz-plane and becomes soft for ? q along ˆ ey. The γ2mode
represents the relative density fluctuations between two
condensates which can also be considered as spin density
fluctuations. This mode remains a free particle spectrum
E(? q). Both of the γ1,2modes, at q ≪ k0, only depend on
the total condensation density NA+ NB = N0. Hence,
the contribution from part I does not lift the degeneracy
between different partitions of (NA,NB) to the quadratic
order of q.
Now, we turn to the low energy excitations in part
II where the dispersion of ψ?k,+is degenerate with the
state of ψ−?k,+but not with ψ2?kA,B−?k,+. The mean field
where ∆N = NA− NB. The Bogoliubov spectra can be
solved as HMF,2=?
The γ1 mode af-
ω(? q) =
z. This is
2(N0− ∆N cosφk)?
NaNbcosφ?keiφ?k+ h.c.?, (5)
ǫ?k(ǫ?k+ gN0) +g2
After summing over?k around the ring, the second term
in Eq. 6 disappears. Thus the ground state zero point
r ? ?
Spin down component
Spin up component
: Spin density vector
x ? ?
down components, and the total density distribution in the
unit of N0 at α = 4 and β = 40 in a harmonic trap. B) The
spin density distribution along the x-axis which spirals in the
z-x plane at an approximate wavevector of 2k0, whose value
at the origin is normalized to 1. The spin density in the whole
plane exhibits the skyrmion configuration.
A) The radial density distribution of spin up and
motion from Eq. 6 selects the optimal partition of ∆N =
0, i.e., an equal partition between two condensates.
Fragmentization of BECs is usually unstable against
perturbations [26, 27].It can be removed by non-
conservation of momentum induced by spatial inhomo-
geneity. Now we consider a realistic system with confin-
ing trap. Our main interest is the interplay between the
confining potential and the Rashba SO coupling which
only lies in the xy-plane. Thus we neglect the effect of
the trapping potential along the z-axis, and assume that
the condensate wavefunction is uniform along this axis.
We consider Vex(r) =
characteristic SO energy scale for the harmonic trap as
Eso= ?λR/l where l =??/(MωT) is the length scale of
the trap; correspondingly, we also define the dimension-
less parameter α = Eso/(?ωT) = lk0. Due to the 2D ro-
tational symmetry, the single particle wavefunctions can
be denoted by the total angular momentum jz= m+1
The ground state single particle wavefunctions form a
Kramer doublet as represented in polar coordinates as
T(x2+ y2), and define the
2(r,φ,z) = Tψ 1
where both f(r) and g(r) are real functions.
Let us gain some intuition about f(r) and g(r) by con-
sidering the strong SO limit α ≫ 1. The Fourier com-
ponents of ψ±1
around the low energy ring. They can be approximately
2in momentum space mainly distribute
dk w(k) e±i1
where the weight w(k) mainly distributes along the low
energy ring with a broadening at the order of 1/l. In
other words, the harmonic potential, which reads Vex=
?A(?k) = i?ψ+(?k)|∂?k|ψ+(?k)?. For the Rashba coupling, the
field strength of?A(?k) corresponds to a π-flux located at
?k = (0,0). Vexbreaks the degeneracy of the plane wave
states with |k| = k0 by quantizing the orbital motion
along the ring as
T(i∂?k−?A(?k))2in momentum representation, where
The ground state forms the Kramer doublets correspond-
ing to m +1
2, and the low energy azimuthal ex-
citations are with m +1
in the radial direction of |k| change the node structures
of f(r) and g(r), which cost the energy at the order of
?ωT and thus are much higher energy states than the
azimuthal excitations at α ≫ 1. For the strong SO cou-
pling α ≫ 1, the single particle ground doublet states
have nearly equal weight in the spin up and down com-
?drdφ r|f(r)|2≈?drdφ r|g(r)|2, thus the
spin moment averages to zero. When the bosons con-
dense into one of the TR doublets, the average orbital an-
gular momentum per particle is ?/2, i.e., one spin compo-
nent stays in the s-state and the other one in the p-state.
This is a half-quantum vortex configuration [28–30], thus
spontaneously breaking TR symmetry.
Next we turn on interactions and define the character-
istic interaction energy scale Eint = gN0/(πl2) and the
dimensionless parameter β = Eint/(?ωT). We numeri-
cally solve the Gross-Pitaevskii equation
2.... The excitations
+ ?(−i∇yσx,αβ+ i∇xσy,αβ) + g(ψ∗
ψβ(r,φ) = Eψα(r,φ)(10)
in the harmonic trap with parameters α = 4 and β = 40.
The interaction effectively weakens the harmonic poten-
tial and does not change the orbital partial wave struc-
ture of the wavefunctions. We show the radial density
profiles of both spin components |f(r)|2and |g(r)|2in
Fig. 2 A. Each of them exhibits oscillations at a pitch
value of approximately 2k0, which originate from the low
energy ring structure and, thus, are analogous to the
Friedel oscillations in fermion systems.
logical spin texture configuration. Let us first look at its
distribution along the x-axis where the supercurrent is
α(r,φ)? σαβψβ(r,φ), an topo-
along the y-direction and the spin lies in the xz-plane.
Explicitly, we express Sz(r,φ) =1
Sx(r,φ) = f(r)g(r). The radial oscillations of |f(r)|2
and |g(r)|2have an approximate π phase shift, which
arises from the different angular symmetries.
result,?S spirals along the x-axis as plotted in Fig. 2 B
at the pitch value of the density oscillations. The spin
density distribution in the whole space can be obtained
through a rotation around the z-axis, which exhibits the
The recent research focus of the “synthetic gauge
fields” in cold atom systems provides a promising method
to observe the above proposed exotic spontaneous half-
quantum vortex [10–12, 14].
have been synthesized by light-atom interaction to gen-
erate vortices experimentally . Similar methods to
generate SO coupling have been proposed [10, 13, 31],
and are currently under experimental implementation. A
convenient method is the “tripod scheme”: three atomic
degenerate hyperfine ground states are coupled to an ex-
cited state by laser fields to create a pair of degener-
ate dark states denoted with pseudospin indices | ↑? and
| ↓?. The spatially dependent laser fields couple the pseu-
dospin and the momentum of the atoms. A typical set of
parameter values are provided in Ref.  for a trap fre-
quency ωT= 2π×10 Hz, a characteristic Rabi frequency
Ω = 2π×107Hz, and a detuning ∆ = 2π×1011Hz, which
satisfy ωT ≪ Ω ≪ ∆ and ωT ≪ Ω2/∆. The value of α
can vary from 0 up to the order of several tens. With the
typical particle number N0= 106, the interaction energy
scale Eintis around 100nK, or 2kHZ , and thus β is
of the order of several tens. Hence, the parameters used
in the Fig. 2 A and B are realistic in real experiments.
The experimental detection would be straightforward as
performed in previous time-of-flight imaging in vortex
experiments [33, 34]. By separately imaging the density
profiles of the pseudospin up and down components, the
radial oscillation of the Sz component can be directly
Another class of boson systems exhibiting SO coupling
is the indirect excitons in 2D coupled double quantum
well systems. We only need to consider the heavy hole
(hh) band with the effective mass m∗
which is separated from the light hole band with a gap of
the order of 10meV. The Hamiltonians of the electron-
hole system reads He= −?2
the Rashba SO coupling strength of the conduction elec-
tron; ε is the dielectric constant; and d is the distance
between the double wells.
of excitons in the BEC limit can be separated from the
electron-hole relative motion. Similarly to Ref. , the
effective Hamiltonian of the 4-component hh excitons de-
noted as (se,jhh) = (±1
2(|f(r)|2− |g(r)|2) and
Artificial magnetic fields
hhand jz = ±3
h, and He−ph= −
|? re−? rhh|2+d2, where
eis the effective mass of the conduction electrons; λRis
The center-of-mass motion
2) can be represented
by the matrix form as
so(?k) Eex(?k) + ∆(?k)
Eex(?k) + ∆(?k) Hso(?k)
where?k is the center-of-mass momentum; M = m∗
is the total mass of the exciton; Eex(?k) = ?2k2/(2M);
Hso(?k) = −m∗
and W(?k) has the d-wave structure as (kx+ iky)2, both
of which are exponentially suppressed by the tunneling
barrier between electron and hole layers, and can be ne-
glected. Consequently, Hexbecomes block-diagonalized.
We consider using circularly polarized light to pump the
exciton of (−1
with heavy hole spin jz=3
the 2D version of the Hamiltonian given by Eq. 1.
In summary, we find that bosons with SO coupling ex-
hibit a novel type of BEC beyond Feynman’s argument.
In the limit of large SO coupling, the degeneracy of the
single particle ground states in momentum space leads to
two fragmented condensations with equal particle num-
ber partition in homogeneous space. In the presence of a
harmonic trap, the condensate spontaneously breaks TR
symmetry and develops the half-quantum vortex config-
uration with the skyrmion type topological spin density
C. W. is supported by NSF under No. DMR- 0804775,
MλR(ky+ikx); ∆(?k) is the exchange integral
2), and then focus on the left-up block
2. We will then end up with
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