Anyonic Excitations in Fast Rotating Bose Gases Revisited

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by  A. Lakhoua

The Role of Anyonic Excitations in Fast Rotating Bose Gases Revisited
A. Lakhouaa, M. Lassauta, T. Massonb, J.C. Walletb
a Groupe de Physique Th´eorique, Institut de Physique Nucl´eaire, F-91406 Orsay CEDEX, France
b Laboratoire de Physique Th´eorique, Bˆatiment 210, 91405 Orsay CEDEX, France
6 The role of anyonic excitations in fast rotating harmonically trapped Bose gases in a fractional
0 QuantumHallstateisexamined. StandardChern-Simonsanyonsaswellas”nonstandard”anyons
0 obtained from a statistical interaction having Maxwell-Chern-Simons dynamics and suitable non
2 minimal coupling to matter are considered. Their respective ability to stabilize attractive Bose

n gases under fast rotation in the thermodynamical limit is studied. Stability can be obtained for
a standardanyonswhilefornonstandardanyons,stabilityrequiresthattherangeofthecorresponding
J statistical interaction does not exceed thetypical wavelength of theatoms.

3
1 PACS numbers: 03.75.Lm, 73.43.-f

]
l
l TheexperimentalrealizationofBose-EinsteinConden- ticular, we find that stability requires that the range of
a
sation (BEC) of atomic gases [1] has given rise to a rich the MCS statistical interaction does not exceed the typ-
h
- variety of phenomena and motivated numerous investi- ical wavelength for the atoms. The description of QHF
s gations focused on ultra-cold atomic Bose gases in ro- within the MCSLG theory stemming from the MCS de-
e
m tating harmonic traps. BEC confined to two dimensions scription of anyons is also discussed.
have been created [2] and their response under rotation Consider an interacting Bose gas at zero temperature
.
t has been studied. Basically, the rotation of a BEC pro- in a rotating harmonic trap with strong confinement in
a
m duces vortices in the condensate [3],[4]. When the rota- the direction of the rotation axis so that the system is
tion frequency increases,the BEC state is destroyedand actuallytwo dimensional. The Hamiltonianin the rotat-
-
d for sufficiently high frequency, a state corresponding to ing frame [14] and the corresponding action quoted here
n Fractional Quantum Hall Effect (FQHE) [5] is expected for further convenience are [15]
o
to possibly occur [6]. In particular, when the rotation
c N
2 [ fhraerqmueonncicycisontfiunniendgtpoottheentcihaalrianctthereisrtaicdifarleqpulaennec,yFoQfHthEe H =AX=121m(pA−A(xA))2+W +V +... (1),
states have been predicted to become possible ground
v
9 states for the system [7]. This observation has been fol-
2 lowed by studies focused on the FQHE for bosons with N m
6 short range interactions [8], [9]. FQHE states involve S0 =Z ( 2 x˙2A−W −V)+Z A(x).J(x,t)+... (2),
8 anyonic excitations [10]. Starting from the standard t AX=1 x
0
05 Cbeheenrns-hSoimwnonisn([C11S])trheaatlizinattiohne tfhorermanoydoynnsam[10ic]a,litlimhiats, pwohteernetiJa(lx,At)=(xP)A=x˙mAω(tǫ)δ(xxj−yxiAel(dts)),ththeeCexotreironliasl gfoarucgee,
at/ atra2cDtivheairnmteornaicctaiollnystcraanppbeedstraobtailtizinegd (BinosaeFgQasHwEitshtataet)- W= NA=1m2i(ωT2A−ω2)|xiAj|2A(2ωπ(resp. ω2πT)istherotation
(respP. trapping)frequency),V= V(x ),V(x )
m thanks to its anyonic (quasi-particle) excitations [12]. A<B AB AB
isthe two-bodypotentialfeltbyPthebosonsintheplane.
- The above mentioned description of anyons is not
d Itcanbe wellapproximatedby V(xAB)=g2δ(xAB)since
unique. Anotherrealizationhasbeeninvestigatedin[13]
n thescatteringbetweenultra-coldbosonsisdominatedby
o where the minimal coupling of the statistical gauge po- the s-wave [14]. Here, g can be viewed as an effective
2
c tential to matter, which is constrained to have Maxwell-
coupling constant encoding basically the effects of the
v: Chern-Simons (MCS) dynamics, is supplemented by a harmonicaxialconfinement(with trappingfrequency ωz
i Pauli-type coupling, as recalled below. While the effec- 2π
X andlocalizationlengthl = 1 )alongtherotationaxis
tiveLandau-Ginzburg(LG)theoriesstemmingfromeach z qmωz
r
anyon realization both reproduce the (so far observed) on the initial 3D scattering properties of the atoms [16].
a
low energy features of the Quantum Hall Fluids (QHF), This will be discussed later on. In (1)-(2), the ellipses
slight differences do exist between CS and MCS anyons. denote possible multi-body interaction terms. We now
Theselatterstemfromastatisticalinteractionhavingfi- consider the ”limit” ω ω for which the trapping and
T
≃
nite range (whereas CS interaction has zero range) and centrifugal potential (nearly) balance each other so that
have an additional attractive mutual interaction (absent W can be neglected.
for CS anyons). In this note, we point out that the exis- The standard CS description of anyons [10]
tence of this attractive mutual interaction may have an obtained from (1)-(2) is achieved by minimally
impact on the ability of (MCS) anyonic excitations to coupling to S a statistical CS gauge potential
0
stabilizeanattractiveBosegasinaFQHEstate. Inpar- a : S S =S + ηǫ aµfνρ+a Jµ where the
µ 0→ 1 0 x 4 µνρ µ
R
2
CS parameter η is dimensionless, Jµ=(ρ;J) with =∂ ia + if0 ; =∂ i(a +A )+ ifi (4a;b).
ρ=ρ(x,t)= δ(x x (t)) and J as given above D0 0− 0 ηe2 Di i− i i ηe2
A − A
(f 1ǫ fPνρ, f =∂ a ∂ a , f (resp. f ) is the
µ≡2 µνρ µν µ ν− ν µ 0 i In (3), φ(=φ(x,t)) is the order parameter, φ φ=ρ, U(φ)
statistical magnetic (resp. electric) field). The equa- †
denotes the LG potential for multi-body interactions,
tions of motion for a take the field-current identity
form fµ=−η1Jµ (i), µinsuring that particle-statistical pUl(inφg)=tegr3m|φs|6h+av..e.(bge3e>n0inacsluinde[1d1i]n).tTheheexntoenndmeidnicmovaalrciaonut-
magnetic flux composite anyonic objects are formed
derivative(4). ThestatisticalMCSgaugepotentialhasa
[10]. Particle-flux coupling responsible for the formation
finite mass[18]M = η e2,sothatthe statisticalinterac-
of the above composite quasi-particles leads to the | |
tion has a finite range Λ = 1 . The CSLG action [17] is
occurrence of Aharonov-Bohm type interactions. The st M
obtained from (3)-(4) by taking the limit e2 . While
anyonic character of the wave functions for the quasi- →∞
CSLG action is believed to encode the (so far observed)
particles is determined by the Aharonov-Bohm phase
[10] exp(i a.dx)=exp i (C is some closed curve) that low energy physics of QHF, it appears that MCSLG ac-
C η tion also reaches this goal and could therefore be used
is inducedRwhen one quasi-particle moves adiabatically
as an alternative description of QHF. This is discussed
around another one, equivalent to a double interchange
more closely at the end of this letter. For the moment,
of identical quasi-particles in the wave function. Then,
we assume that the system is in a FQHE state (η=0)
the statistics of the quasi-particles is controlled by 6
described the MCSLG action (3)-(4).
the value of η: they have Fermi(resp. Bose) statistics
whenever η= 1 (resp. 1 ), k Z. The statistics The equations of motion derived from (3) are
(2k+1)2π (2k)2π ∈
is anyonic otherwise. Notice that, when η= 1 , 1 δU(φ)
(2k+1)2π i φ+ φ 2g (φ φ)φ= (5a),
theinitialBosestatisticsoftheatomsisconvertedintoa D0 2mDiDi − 2 † δφ
†
Fermi statistics for the quasi-particle which corresponds
to the so called statistical transmutation [10]. This,
1 1 1
plugged into a second quantized formalism, leads to ǫ ∂µ(fρ+ ρ)+(f + )=0 (5b),
−e2 αµρ ηJ α ηJα
the usual CSLG action [17] underlying the main part
of the analysis presented in [11]. Since the statistical where =ρ, = i (φ φ ( φ) φ)and isdefined
interaction is mediated by a CS gauge potential, its J0 Ji 2m †Di − Di † Dµ
in (4) while anyonic configurations,on which we now fo-
range is zero. cus, are obtained from (5b) when f = 1 [19]. From
An alternative realization for anyons has been µ −ηJµ
the field’s conjugate momenta Π =Π = 0, Π =iφ ,
proposed and discussed in [13]. It is obtained φ† a0 φ †
Π =1 f ηǫ aj+ 1 ǫ j, one obtains the Hamilto-
by coupling a MCS statistical gauge potential aµ ai e2 0i−2 ij ηe2 ijJ
nian
to S through minimal and non minimal (Pauli-
0
type) coupling term, the strength of this latter 1 1 1
H = (f + ρ)2+ Θf2
=beSin+g fixed1 tfo afµsνp+ecηifiǫc vaaµlufeνρ+[1a3],Jnµam1elyf: JSµ0→(Se22 Zx 2e2 0 η 2e2 i
0 x−4e2 µν 4 µνρ µ −ηe2 µ
has mRass dimension 1), where the coupling constant
1 1
of the Pauli-term (last term in S2) has been already +2m|Diφ|2+(g2− 2η2e2)ρ2+U(φ) (6),
fixed to the above mentioned specific value. Then,
−thee12ǫeαqµuρa∂tµi(ofnρs+ofη1Jmρo)+tioηn(fαfo+r η1aJµα)c=an0 wbehicwhriisttseonlveads TwhheerepoDsiit=iv∂iit−yi(oafi +theAim) oadnedl wreeqdueirfiensedρ Θm=η12−e2mη[ρ22e02]..
by(i). Thislatterobservationhasbeenusedasastarting Restoring ~ and c, this translates into ρ≤ρ , ρ =
lim lim
point to construct an effective theory reproducing the η ( Λ ) 1 where Λ= ~ is a typical (de B≤roglie) wave-
usual anyonic behavior [13]. The Aharonov-Bohmphase l|en|gΛtsht foΛr2the atoms amncd Λ = ~ (e2=µ˜c2). This con-
definingthe statisticsofthe resultingquasi-particlesstill st ηµ˜c
dition shouldbe fulfilled by cur|re|ntexperimental values,
verifies exp(i a.dx)=exp i so that η again controls
C η providedΛ < (Λ),aconditionthatwenowassume[21].
the statistics.R However, anyons obtained through this Namely, fosrt∼7OLi, 23Na, 87Rb, one obtains respectively
construction have an additional attractive contact
ρ η ( Λ ) 1029, 1030, 1031 cm 2, which, for possibly
mutual interaction (not present in the CS case) [13]. lim≃| | Λst −
reachable current Quantum Hall states, should be larger
Combining S with the second quantization machinery,
2 by several orders of magnitude than the experimental
one obtains the corresponding MCSLG action
values reached by the matter density so that ρ 1.
mη2e2≪
1 Compared to the CSLG Hamiltonian [11], H involves
S = iφ φ φ2 g (φ φ)2 U(φ)
Z †D0 − 2m|Di | − 2 † − additional contributions from the Maxwell part of (3)
x
and the non minimal coupling terms in (4). The inter-
−41e2fµνfµν + η4ǫµνρaµfνρ (3), tahcetioPnauenlie-trygpyeg2cρo2uprelicnegiviens(c4oan)t:ritbhuetsioenlsatftreormco2m1e2bfi02neantdo
3
yield the first term in (6) together with the additional These latter can be parametrizedas ρ(z)=4η h(z)2(1+
′
adsotetmpreaenhcdtoiiwvnega(omltneartghnteehterieclc)aoctniovcnelturmsiiboaungtniooitbnutd−aein2oηeρ2fd2eΛ2intaon[g1d21ρΛ]2sf.tr,oTmmhaisya, Nh|h(,(zzg))i.v|2eT)s−hr2eispe(aztr=otixcr1au+dlaiiarxl2clyh)osifycomermha(enzty)r=ich(vzo0olo/rmtze)oxnr-tp(yzhp=|icersefoiuθlnu)|,ctitnoion∈ns
CSLG descriptionof anyonsabout the ability of statisti- ρ(r)=4ηrn22((rr0)n +(rr0)n)−2 where r0 is some arbitrary
cal interaction to stabilize an attractive Bose gas, as we length scale and φ=√ρeinθ.
now show.
Wesetφ=γeiϕ √ρeiϕ,aˆi=∂iϕ (ai+Ai),η>0. ForN Let us discuss the above analysis. From (7)-(9), one
atoms in the trap≡, one has φ φ−=N. Using f = 1 realizes that the initial two-body coupling term g2 can
x † µ −ηJµ be compensatedby the statisticalinteraction. In∼view of
andassumingthatfieldsvanRishatinfinitysothatbound-
(7), for fixed η>0, the particular value g (8) represents
arytermsdisappear,thestaticenergystemmingfrom(6) 0
the limiting value for the two-body coupling constant g
can be conveniently written as 2
below which the Bose gas cannot be stabilized by the
H =ωN + 1 (∂ γ ǫ aˆjγ)2+(g g )ρ2 statistical interaction. In the CS case, Λst=0 and g0 is
Zx 2m i − ij√Θ 2− 0 negative. Then, when g2=g0, corresponding to an initial
attractive Bose gas, the system behaves as a free anyon
gas whose ground state (neglecting the small φ6 and
| |
∞ ρ higher order terms in U(φ)) is exactly described by (10)
+ mη2e2 (ηe2C +2ωC )( )k+U(γ) (7).
Zx X k−1 k mη2e2 giving rise to non singular finite energy matter distribu-
k=3 tion [22]. Then, one would conclude as in [11] that an
attractive Bose gas may be stabilized against collapse in
In (7), ω>0, the positive constants C ’s are given by
k
(1−mηρ2e2)12=1− ∞k=1Ck(mηρ2e2)k (Ck=Γ2(kk!−√12π) whereΓ tmhiengthferrommoadyCnSam(ziecraolrliamnigte)bystaatniysotinciacleinxtceitraatcitoionns.stTehmis-
is the Euler functPion) and we have defined
conclusion is somehow altered in the case of ”non stan-
dard”MCSanyons. Inthiscase,g receivesanadditional
~2 Λ ~ω ~2 Λ 0
g = (1 st(1 )) (1 st) (8), positive contribution from the (statistical) magnetic en-
0 −2mη − Λ −2mc2 ≃−2mη − Λ
ergy and the magnetic coupling in (4a) and the sign of
where ~ and c have been reinstalled and the rightmost g0 depends now on the relative magnitude of Λ and Λst
relation stems from ~ω mc2 which holds for current (see (8)). When g2=g0, the system behaves again as a
experimental values for ω≪(taking ω (10)- (103) Hz freeMCSanyongaswhosegroundstate(neglectingagain
2π∼O O small γ6 and higher order terms) is now accurately de-
as a benchmark). In the same way, the term ωC in
k
(7) can also been neglected (since ~ω ηµ˜c2 in∼view of scribed by (10). However, this situation can be reached
Λ < (Λ)). ≪ by an initial attractive Bose gas only if Λst<Λ corre-
st
T∼hOe quartic interactionterms in (7)canbe eliminated sponding to negative values for g0. A MCS statistical
interaction with range exceeding the typical wavelength
provided g is chosen to be
2
Λ for the atoms couldnot protectanattractive Bosegas
from collapse (in the thermodynamical limit).
g =g (9).
2 0
Now, let us assume that Λ <Λ and discuss more
st
Then, neglecting for the moment in (7) the small terms
closely (9) from a physical viewpoint. In (most of) cur-
of order (γ6) (and higher) as in [11], the ground
∼O rent experiments, 2D atomic systems are obtained by
state of (7) is obtained for those configurations satisfy-
tightening the axial confinement applied to the initial
ing (∂ γ ǫ aˆjγ)=0, i=1,2. This, combined with (5a),
i ij 1 (harmonically trapped) 3D system, which restricts the
− Θ2
(8), (9), further expanding the various contributions de- dynamics of the atoms along the axial direction to zero
pending on Θ in powers of ρ and using the fact point oscillations [16]. Kinematically, the system be-
mη2e2
that ρ 1 yields ∂2γ (∂iγ)2= γ(γ2(1+ ω )+2mω) comes2Dwhilethe effectivecouplingconstantg2 forthe
wherem,ηi2ne2t≪he RHS, ωi −1 stγill ho−lds sηo thatηet2his latter interparticle interaction depends closely on the motion
ηe2≪ of the atoms in the axial direction [16]. As shown in
equation can be accurately approximated by
[16], g =4π~2(√2π(lz)+ln(B~ωz)) 1 (B 0.915,ǫ is the
2 m as π ǫ − ≃
(∂ γ)2 γ2 typical energy for the relative motion of the atoms and
∂i2γ− iγ =−γ( η +2mω) (10). as is the (3D) s-wave scattering length) where ǫ ~ωz
≪
must hold. This regime (called quasi-2D regime in [16]),
When ω goes to zero, the solutions of (10) reduces relevant here, seems to be accessible to experiments [2],
smoothly [22] to the solutions of the Liouville equation [16]. Then,(9)holdsprovideda istunedtothenegative
s
value a = √2πl ( 8πη +ln(B~ωz)) 1. Typical values
∆lnρ=−η2ρ (11). obtainesd f−or ~ωǫz∼zO1(−1Λ0Λs2t−103)πarǫe |alz−s|∼O(10−1) (resp.
4
|alzs|∼O(10−1−10−2)) for a CS (resp. MCS) statistical Θ0=mnη20e2 (ω02(p)≃ 4pm42+p2mµ +mn220η2 since Θ0≪1).
interaction (assuming possibly reachable current Quan- This indicates that the system has a gap, so that the
tumHallstatesandΛst<12Λ). Letusdiscussthepossible fluid described by S′ is incompressible.
effect of the inclusion o∼f the next to leading order inter-
actionterms φ6 onthestabilityofthesystem[23],as- Although CS and MCS anyons share many similar
∼| |
suming that (9) holds. In the CS case, stability requires properties (in particular those related to the statistical
g >0,asindicatedin[11]. IntheMCScase,the φ6term phase), MCS anyons feel an additional attractive con-
3
| |
inU(φ)receivesanadditionalpositive (repulsive)contri- tact mutual interaction. Therefore (when Λst=0), CS
6
bution coming from the infinite sum in (7). Then, sta- andMCSanyonsmustbeconsideredasdistinctcompos-
bility in the MCS case is obtained provided g + 1 ite objects, both carrying anyonic statistics. This can
3 8m2η3e2
>0. The actualcomputation of g , which here must cor- be understood from the mechanism [13] responsible for
3
respondtothequasi-2Dregime,isstilllackingandwould the formation of each type of these particle-flux anyonic
need to extend the analysis reported in [16] to the case composite (quasi-particles), reflecting itself in each case
of higher order interactions within the quasi-2D regime. intotheoriginofthefield-currentidentityi). Recallthat
QHF, usually described in the low energy regime by i) physically expresses the fact that for a distribution of
a CSLG theory [17], can alternatively be described by a point-like particles only localized (statistical) fields can
MCSLG theory as defined in (3)-(4). To compare to the appear. ThisisautomaticallyfulfilledintheCScase(see
CS case [17], it is convenientto add a chemicalpotential S1)sincetheCSstatisticalinteractionhaszerorange. In
termto(3),S =S+ µφ φ(µ>0)andsetU(φ)=0inall. the MCS case for which the interaction has finite range,
′ x †
(5a)becomesi φ+R1 φ 2g (φ φ)φ=µφwhilethe i)resultsfromanexactcancellationbetweenelectricand
relevant anyonDic0con2fimgDuriaDtiion−s ar2e s†till obtained when Pauli-typecouplingeffects[13],whichamountstofixthe
f = 1 solving (5b). Then, the equations of motion Paulicouplingconstanttoaspecificvalue(see[24]). This
µ −ηJµ leads to the appearance of an additional attractive con-
admittheuniform(constant)densitysolutionminimizing
tact interaction (whose strength grows as Λ increases)
the energy H =H µρ, given by st
′ − x amongtheresultingMCSanyonswhich,apartfromthis,
R
µ
φ=√n ; a+A=0; a =0; n = (12a;b;c;d), behave essentially as CS anyons. The relevant question
0 0 0
2gˆ
2 is now to examine if some effects specific to the MCS
where gˆ=g 1 , similar to the usual uniform den- framework, in particular those related to the above con-
2 2− 2η2e2 tactattraction,maybeexperimentallyobserved. Clearly,
sitysolutionsupportedbytheCSLGactionforQHF[17].
the “electronic” Quantum Hall systems are excluded:
(12b)impliesthattheexternalmagneticfieldisscreened
the fact that the repulsive Coulomb interaction among
by the statistical magnetic field. This combined with
f = 1ρ, yields ν=2πη where ν is the filling factor. electronscannotbe manipulatedcombinedwith the spe-
0 −η cificfeaturesoftheexperimentallyaccessibleobservables
When statistical transmutation occurs as expected in
Hall systems, ν= 1 , k 0. Introducing polar coordi- make these systems only sensitive to the properties both
2k+1 ≥ shared by CS and MCS anyons in the long wavelength
nates (r,θ) and setting φ=√ρeinθ, a=eθα(rr), a0=a0(r) limit. Infastrotating(harmonicallytrapped)Bosegases,
(e =(sinθ, cosθ), n Z is the winding number), one
θ − ∈ the interaction among atoms can be manipulated, offer-
finds that H admits static finite energy vortexsolutions
′ ingawaytostudythe stabilizationofinitiallyattractive
satisfying the boundary conditions ρ n , α(r) mωr2
∼ 0 ∼ Bose gases in a FQHE state through their anyonic exci-
for r , ρ 0, α(r) n for r 0. The vortex ef-
→∞ → →− → tations. Aswehaveshown,stabilitymaybeinparticular
fective magnetic flux Φ (f +F ) is then Φ= 2πn
(F =ǫ ∂iAj). Using n≡=RxηF0, f0= ρ, Φ is re−lated somehow conditioned by the actual nature of the statis-
0 ij 0 0 0 −η tical interaction. Basically, anyons of MCS origin loose
formally to the ”vortex charge” Q= x(ρ−n0) through their ability to stabilize the system as the range of the
Q= ηΦ=2πηn. When statistical tranRsmutation occurs, MCSinteractiongrows. Wenotethataninterestingpro-
Q=−2kn+1 so that the vortices can be interpreted as (the posaltomeasurethestatisticalphaseofanyonshasbeen
analogue of) the fractionally charged Laughlin quasi- presented in [25]. This, combined with an experimen-
particles, as it is the case in the CS description of QHF tal implementation of the analysis presented here may
[17]. Finally, by expanding S′ around the (mean field) provide a deeper insight into the physicalfeatures of the
uniform solution (12) up to the quadratic order in the anyonic composite quasi-particles.
fields fluctuations, fixing the gauge freedom, integrating
out the fluctuations of the statistical field, one obtains
the low energy effective action for the matter degree of
freedom. It yields the following dispersion relation
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thesystemisformally equivalenttothatof2D(bosonic)
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