Quantum Monte-Carlo method without negative-sign problem for two-dimensional electron systems under strong magnetic fields

0.21 MB·English

by  Sei Suzuki

Typeset with jpsj2.cls <ver.1.2> Letter
Quantum Monte-Carlo method without negative-sign problem
for two-dimensional electron systems under strong magnetic fields
Sei Suzuki∗ and Tatsuya Nakajima1
4
0
Department of Basic Science, University of Tokyo, Tokyo 153-8902
0
2 1Department of Physics, Tohoku University, Sendai 980-8578
n (ReceivedFebruary2,2008)
a The quantum Monte-Carlo method is applied to two-dimensional electron systems under
J
strong magnetic fields. The negative-sign problem involved by this method can be avoided

3 for certain filling factors by modifying interaction parameters from those of the Coulomb in-
2 teraction. Our techniques for obtaining sign-problem-free parameters are described in detail.

Calculated results on static observables are also reported for Landau level filling ν =1/3.
]
l
l KEYWORDS: quantumMonte-Carlo,negative-signproblem,linearprogramming,fractionalquantumHall
a
h states
-
s
e
m It has been revealed that a two-dimensional electron QMCmethodisaccompaniedbythenegative-signprob-
system in a strong magnetic field has various ground- lem, which sometimes makes us away from investigat-
.
t state phases. In particular, the existence of incompress- ing physicalproperties.Thus wefirstneedto resolvethe
a
m ible liquids gives rise to the fractional quantum Hall ef- negative-sign problem in order to make the best use of
fect1–3 and has attracted much theoretical and experi- the present method. In this letter, we present one solu-
-
d mentalintereststothisso-calledquantumHall(QH)sys- tion to the negative-sign problem in the QH system.
n tem.RichquantumphasesrealizedintheQHsystemare This letter is organized as follows. We first express
o
attributed to Coulomb interactions between electrons, physical quantities in the QH system in terms of the
c
because the kinetic energy is quenched by the Landau auxiliary-field path-integral. Then we describe how to
[
quantizationinastrongmagneticfield.Thustheground avoid the negative-sign problem in the present system.

1
state involves strong electronic correlations and is diffi- Finally some results of QMC calculations are presented.
v
cult to consider by use of mean field theory. The system studied here is composed of interacting
6
4 Sofarseveralmany-electronstateshavebeenproposed electronsconfinedonasphericalsurface.12 Itis assumed
4 theoretically for the ground state in each phase. The that a magnetic monopole is located at the center of
1 Laughlinstate,4 Jainstate,5 and Pfaffianstate6 success- the sphere. The monopole induces a uniform magnetic
0
fully account for incompressible liquid phases in the QH field on the surface. The number of flux quanta diverg-
4
system.Ontheotherhand,thecomposite-fermionliquid ing from the monopole is denoted by 2s hereafter (2s:
0
/ states7,8 andanisotropiccharge-density-wavestates9 are integer). Single-electron states on the sphere are spec-
t
a plausible to compressible phases at the half filling of the ified by the Landau level index and the z-component
m lowest and higher Landau levels, respectively. quantum number, m, of angular momentum (AM). We
- When the properties of such strongly-interacting sys- consider the strong magnetic-field limit and restrict the
d
tems are investigated, numerical studies can provide a single-electron states to the subspace of the lowest Lan-
n
variety of significant information. The exact diagonal- dau level (LLL). Then a single-electronstate is specified
o
c ization method of the Hamiltonian matrix has been fre- byonlymwhichrangesfrom stos,andthenumberof
−
: quently usedto inspect the variouspropertiesofthe QH degeneratesingle-electronstatesamountstoN =2s+1.
v s
system10 so far, but its applicability is limited to small- The spin degreesof freedomare neglectedfor simplicity.
i
X size systems. Recently the density matrix renormaliza- In the LLL approximation, the kinetic energy is con-
r tion group (DMRG) method has been applied to the stant for QH systems containing the fixed number of
a
studyoftheground-statepropertiesoftheQHsystem.11 electrons.TheremainingtermoftheHamiltoniancomes
ThesystemsizepermittedintheDMRGmethodismuch from Coulomb interactions between electrons. We ex-
largerthanthatintheexactdiagonalizationstudy,butit press the interaction Hamiltonian in a bilinear form
is difficult to study dynamical quantities by the DMRG of the density operator to introduce the Hubbard-
method. Stratonovich transformation needed for QMC calcula-
In the presentstudy, we propose an application of the tions.
quantum Monte-Carlo (QMC) method to the QH sys- The density operator is defined using
tem. This is because the QMC method can investigate the Clebsch-Gordan coefficient as ρ =
KN
not only static but also dynamical quantities in large- s s KN sm ;sm a† a˜ , where a
m1=−s m2=−sh | 1 2i m1 m2 m
size systems. This method can be applied for both zero- iPs the anPnihilation operator of an electron with AM
and finite-temperature cases and has ever been used for quantum number m and a˜ = ( 1)s+ma is the time
m −m
−
a variety of quantum many-body systems. However, the reversal of a . The interaction Hamiltonian is written
m
∗E-mailaddress:[e-mail]
2 J.Phys.Soc.Jpn. Letter SeiSuzukiandTatsuyaNakajima
in terms of the density operator as ζ(σ)/ζ(σ) isthesignofζ(σ).Iftheexpectationvalueof
| |
ξ(σ)isvanishinglysmall,Monte-Carloevaluationsof
1 2s K hOi
= χ ( 1)K+Nρ ρ +ǫ ρ . (1) becomeveryunstable.Thisisthe so-callednegative-sign
K KN K−N 0 00
H −2KX=0 NX=−K − problem. Thus, in order to make a precise evaluation, it
is desirable that ζ(σ) is always positive.
Here ǫ0 = − 2Js=0(−1)2s−J(2J +1)VJ/(2√2s+1) and For detailed discussion on the negative-sign prob-
the coefficienPt χK is given in terms of the Haldane lem, let us move to the matrix representation of
pseudopotential,13 VJ, as χK = 2Js=0TKJVJ. The ma- the QMC method.14 In order to exploit the time-
trix T is explicitly written usingPWigner’s 6j symbol as reversal symmetry, we first define fermionic opera-
TKTJh=e Pint2Jes=ra0c(−tio1n)2sH−aJm+Kilt+o1n(i2aJn+ca1n){asssslJKso}.be written in t(or1s),s−αmma, byforαmm <=0.aBmy ifnotrromduci≥ng 0a,maantdrixαMm(σ=)
m
awnhsmoetr1he=e−rAsJfoMrsmmi2s=a−tshsheHJpMa=i|rsimn12g1P;osmp2Jse=2r0iaaVtmoJr1PagmJMiv2e=.n−SiJbnAyce†JAMaJmAMJsaM=t-, aiUn−s(βt−h;0∆e)βca≡hn(σoen)Mic(a=σl(NetPn))s·esm·m,·nbe=Ml−e(sσiM(s1)m)p.nro(Tσdh)ueαce†mwdαeinbg,yhtwoepfuenrdacettfiiionnnge
iPsfies thePanti-commutation relation, AJM = 0 for even the particle-number projection to that in the ground-
2s J. Thus the terms with V for even 2s J are un-
− J − canonical ensemble:15
physical and have no influence on physical quantities.
Thereforetheseunphysicalpseudopotentialscanbeused Ns
ζ(σ)= e−2πiNe(I/Ns)det[1+e2πi(I/Ns)U(β;0)], (3)
as tunable parametersto improvethe efficiency of QMC
calculations. XI=1
Theimaginary-timeevolutionoperatore−βHisdecom- where Ne is the number of electrons. In the zero-
posedto the productofimaginary-timeslicese−∆βH (β: temperature formalism, the weight function is given by
inverse temperature, ∆β β/N , N : Trotter number). ζ(σ) = det[V†U(β;0)V], where an arbitrary state is
t t
By introducing the Hubb≡ard-Stratonovich transforma- written in terms of a matrix V and electron vacuum 0
tion for each slice, the evolution operator e−βH can be as|ψi= sm=−sα†mVm1 ··· sm=−sα†mVmNe |0i.W| ei
written in terms of a linearized Hamiltonian given by remark th(cid:0)Pat M and U ar(cid:1)e squ(cid:0)aPre matrices of di(cid:1)mension
N , while V is a N N rectangular matrix.
h(σ) = 1 2s η χ K 2(σ∗ ρ +σ ρ˜ ) sFor the zero-tempse×ratuereformalism,the negative-sign
2 K K(cid:20) KN KN KN KN problem is overcome under the following conditions:16
KX=1 NX=1
(i) 2s is odd and N is even.
e
1
+σ∗ ρ +σ ρ˜ + χ (ρ )2+ǫ ρ .(2) (ii) χK is non-negative for K =1,2, ,2s.
K0 K0 K0 K0(cid:21) 2 0 00 0 00 By the condition (i), the dimension·,·N· , of matrix M
s
Here ρ˜ = ( 1)Kρ† , σ is an auxiliary field for becomes even. The condition (ii) yields ηK = 1 in
amodeK(NK,N)−introduKcNedbKyNthe Hubbard-Stratonovich Eq.(2) and then M satisfies M−m1,−m2 = Mm∗1,m2 and
M = M∗ for positive m and m . There-
transformation,andηK isanumericalfactorwhichtakes for−em, 1b,ym2arra−nginmg1,A−Mm2indices, m, in t1he matr2ix repre-
1fornon-negativeχ andifornegativeone.Weremark
K sentation as m = s, ,1/2, s, , 1/2, M has such
thatthe terms,χ00(ρ00)2/2andǫ0ρ00,inEq.(2)arecon- a form as ··· − ··· −
stantsbecauseρ isproportionaltothefixednumberof
00
electrons. A B
M= , (4)
The partition functions, Z = Tr[e−βH] for canoni- (cid:20) B∗ A∗(cid:21)
−
cal ensemble and Z = ψ e−βH ψ for zero tempera- where A and B are square block matrices of dimension
h | | i
ture, are represented in terms of auxiliary-field path-
N /2. When matrices M and M are of the form of
s 1 2
integral as Z = σζ(σ). Here the weight func-
Eq.(4), the product M M is also of the form of Eq.(4).
D 1 2
tion is defined by Rζ(σ) Tr[U(β;0)] for canonical Thus U(β;0) also has the form of Eq.(4). When we
≡
ensemble and by ζ(σ) ψ U(β;0)ψ for zero tem- chooseamatrixV thatsatisfiesV =V∗ and
≡ h | | i −m,Ne/2+n m,n
perature, U(β;0) in the weight function is defined by V = V∗ for positive m and 1 n N /2,
U(β;0) e−∆βh(σ(Nt)) e−∆βh(σ(1)), and ψ in the th−emf,onllow−ingmp,Nroe/p2e+rnty is also obtained: ≤ ≤ e
≡ ··· | i
zero-temperature formalism is an arbitrary state not or-
P Q
thogonaltothetruegroundstate.Theexpectationvalue V†U(β;0)V = , (5)
(cid:20) Q∗ P∗(cid:21)
of an observable is written in the path-integral rep- −
O
resentation as = σ ζ(σ)/Z, where where P and Q are square block matrices of dimension
σ σ
hOi D hOi hOi ≡
Tr[OU(β;0)]/ζ(σ) for Rcanonical ensemble and hOiσ = Ne/2. Then, if λi is an eigenvalue of V†U(β;0)V, its
ψ U(β;β/2) U(β/2;0)ψ /ζ(σ) for zero temperature. complex-conjugate is also shown to be its eigenvalue.
h | O | i
Theauxiliary-fieldpath-integralisevaluatedbymeans Thus we obtain ζ(σ)= Ne/2λ λ∗ 0.
i=1 i i ≥
of the Metropolis-Monte-Carlo technique. However, the We can avoidthe negaQtive-signdisaster alsoin case of
weight function, ζ(σ), can be negative for certain the canonical ensemble by realizing the following condi-
configurations of auxiliary fields. Thus let us regard tions: (i′) 2s is odd, and (ii) χ 0 for K = 1, ,2s.
K
≥ ···
P(σ) ζ(σ) as a positive-definite probability distri- WenotethatevenN isnotrequiredin(i′).Sincethema-
≡ | | e
bution. The expectation value is then written as = trixMandU(β;0)thenhavetheformofEq.(4)asinthe
hOi
σ[ σξ(σ)]P(σ)/ σξ(σ)P(σ), where ξ(σ) zero-temperature formalism, the weight function in the
D hOi D ≡
R R
J.Phys.Soc.Jpn. Letter SeiSuzukiandTatsuyaNakajima 3
ground-canonical ensemble, det[1+U(β;0)], always be- ]
B
comesnon-negative.Theweightfunctioninthecanonical el /2
ennusmembebrlep,rwojheiccthioins toobttahiantedofbtyheopgerroautnindg-ctahneonpiacratlicelne-- V [J Coulomb
model I
semble[see Eq.(3)],is expectedto be almostalwayspos-
model II
itive.Wehavenumericallyconfirmedthatthisisthecase.
02468024680
The next task is to inquire which systems satisfy the 1J11112
negative-sign-free conditions. The condition (i) [(i′) for
the canonical ensemble] restricts the Landau-level fill-
ing accessible by the QMC method. For instance, the Fig. 1. Pseudopotentials for negative-sign-free interactions and
the Coulomb interaction within the lowest Landau level with
ν = 1/m Laughlin state (m: odd), characterized by
Ns=2s+1=22.Onlyphysicalcomponentsforodd2s−J are
2s = m(N 1), can be studied by the QMC method
e shown. The pseudopotentials of negative-sign-free model inter-
−
ifNe iseven.Thecondition(ii)bringsaboutstrongcon- actionnamedIareobtainedbythelinearprogrammingmethod
straints for electron-electron interactions accessible by and are shown by cross symbols. The pseudopotentials of in-
the QMC method. In fact, this condition is not satisfied teraction II shown by closed triangles are obtained by applying
uniformshiftsandrescalingtothoseofinteractionI.
by the Coulomb interaction within the LLL.
As mentioned above, the Haldane pseudopotentials,
V , are classified into the physical components (2s
JJ: odd) or unphysical ones (2s J: even). We ob−- method, we used the following ones: V2s−1 ≥V2Cs−1 (the
tain coupling constants χ favor−able to QMC calcu- shortest-rangecomponent) andVJ ≤VJC for J 6=2s−1.
K This is because the Laughlin state favors prominently
lations by using the unphysical ones as tunable pa-
largeshort-rangecomponents.Pseudopotentialsofmodel
rameters, with minimizing the difference of the phys-
interactionI agreewellwith those ofthe Coulombinter-
ical ones between our model interaction and the
action on short-range components, but do not on long-
Coulomb interaction. Then the model pseudopoten-
range ones. In fact, the long-range components shift
tials are given in terms of obtained χ by V =
K J downward from those of the Coulomb interaction and
2Ks=0TJKχK, because KTJKTKJ′ =δJJ′. We define eventually become even negative.
tPhe quantity to be minimPized by f(−χ0,χ1,··· ,χ2s) = InordertomakeimprovementsontheinteractionI,we
Ppse2us−dJo:pooddteκnJtia(cid:16)lPof2Kst=h0eTCJoKuχloKm−bVinJCte(cid:17)r,acwtihoenreanVdJCκiscothne- p(ie)rufonrimfortmhesfhoilfltowofinpgseoupdeorpaotitoennstioanls,itVspseuVdop+otVenIt,ia(ilis):
J J → J | 0|
trolsthepriorityofeachpseudopotentialcomponent.We energy rescale, V V (VC VC)/(VI VI),
J → J × 2s−1 − 0 2s−1 − 0
note that the arguments of f should be non-negative, and(iii)anotheruniformshiftofpseudopotentials,V
J
→
which agrees with the condition (ii). Although the first V + VC. As a result of these sequential operations,
J 0
argument, χ , is not referred by the condition (ii), it we obtain another interaction named II, whose pseu-
0
−
must be positive in order to make a repulsive pseudopo- dopotentials are given by VII = VC + (VI VI)
tentialV = 2s T χ >0undertheconditionsthat (VC VC)/(VI VI). WJ e note0here thJat−uni0form×
0 K=0 0K K 2s−1 − 0 2s−1− 0
χK > 0 for P1 K 2s and T0K < 0 for all K. Such shifts of pseudopotentials make no influence on the elec-
≤ ≤
a set of χ can be obtained by the linear programming tronic states, because the change, V V +v for all
K J J
→
method, which minimizes f over its non-negative argu- J, involves just a constant term in the Hamiltonian as
ments imposing the inequalities V VC (V VC) for 1v A† A = 1vN (N 1). Furthermore these
J ≥ J J ≤ J 2 JM JM JM 2 e e−
non-negative (non-positive) κJ. opePrations do not break the negative-sign-free character
Asimpleconditionforthelinearprogrammingmethod ofχ ,becauseuniformshifts ofV bringaboutonlythe
K J
is given by κ = 1 for all of odd 2s J. Then physi- changeinχ andenergyrescalekeepsthe signofχ .
J K=0 K
−
cal pseudopotentials are determined with equal weights Thus the interaction II is also free of negative-sign.
and become larger than those of the Coulomb interac- The pseudopotentials of interaction II are shown by
tion, respectively. The coupling constants χ obtained closed triangles in Fig.1. The energy scales of interac-
K
in this way vanish except for the cases of K = 0 and tion II and Coulomb interaction are the same, because
K = 1. The Hamiltonian for this interaction is written V V gives the energy scale. The character of in-
2s−1 0
−
asH = 1χ N2/N +6χ 2/[(N 1)N (N +1)],where teraction II is similar to that of the Coulomb interac-
−2 0 e s 1L s− s s
denotes the total AM operator. The eigen-energies of tiononshort-andlong-rangecomponents,while the dif-
L
this Hamiltonian depends only on the total AM quan- ference between pseudopotentials of the interaction II
tum number. Therefore the groundstate is energetically andCoulombinteractionisnotsosmallonmiddle-range
degenerate and the excitation spectrum is gapless. ones.Inspiteofthis disagreementonmiddle-rangeones,
Figure 1 shows the pseudopotentials of the Coulomb it is shown later that the interaction II leads to the
interaction and two negative-sign-free interactions. For ground-state properties qualitatively similar to those of
the interaction I whose pseudopotentials are shown by the Coulomb-interacting system.
crosssymbols,wesetthepriorityparametersasκ = 1 Figure 2 shows the results of zero-temperature QMC
0
−
and κ = 0 for J = 0. Namely we minimized the differ- calculationsforthe QHsystemwith2s=21andN =8
J e
6
ence, V VC ,betweenthelongest-rangecomponentsto (ν = 1/3). These calculations were done by using the
| 0− 0 |
mimicthelong-rangenatureoftheCoulombinteraction. negative-sign-free interaction II in Fig.1. Calculated re-
As the inequalities imposed in the linear programming sults for three imaginary-time slice widths, ∆β = 1/4,
4 J.Phys.Soc.Jpn. Letter SeiSuzukiandTatsuyaNakajima
el2 /]B ∆∆ββ == 00..12255 ∆∆ββ == 00..12255 tthroenawnnitihhiltahtrieoenfloupxerqautoarntoaf aatctoamchpeodsittoeibt)o.soFnig(uarnee3l(ebc)-
[
cle ∆β = 0.0625 ∆β = 0.0625 shows C3(r) for β = 8 and 64 [1/(e2/lB)] obtained by
rti (a) (b) QMC calculations as well as that at T = 0 obtained by
pa L2| theexactdiagonalization.Atzerotemperature,C3(r)re-
er | mains finite for large separation r. This result suggests
p
y that the ground state for the interaction II should have
g
er theODLRO.Forfinitetemperaturesβ .8,theODLRO
n
E correlation rapidly vanishes with increasing separation
00000000 00000000 r, while it survives at β = 64 6/∆E as in case of
34567890 34567890 ≃
β [1/(e 2/lB1)] β [1/(e 2/lB)1] T =0.Thisisbecausethecontributionsofexcitedstates
to C (r) is negligible for temperatures quite lower than
3
∆E.
Fig. 2. Expectationvaluesof(a)energyperparticleand(b)total We comment on the validity of negative-sign-free in-
angular momentum by the zero-temperature QMC calculations
teraction II as a substitute for the Coulomb interaction.
of the QH system with 2s = 21 and Ne = 8 (ν = 1/3). The
unitof energy ise2/lB (lB: themagnetic length), β the inverse By the exactdiagonalizationmethod, weconfirmedthat
temperature, and ∆β is the width of imaginary-time slice. The the overlapbetween the two ground states for the inter-
energy contribution by the positive-charge background is also action II and Coulomb interaction is not large ( 0.35)
takenintoaccount. for ν = 1/3. However, the existence of the ODL∼RO in
the ground state and that of energy gap above it sug-
gest that an imperfect Bose condensation of composite
1/8, and 1/16 [1/(e2/l )] are almost the same. Hence
B bosonsexistsforinteractionII,althoughitisnotsoper-
the numerical errors introduced by the Suzuki-Trotter
fect as that in the Laughlin state. Thus we believe that
approximation are not so important for ∆β . 1/4. For
the ground state for interaction II contains the essential
the inverse temperature β & 60/(e2/l ), both expecta-
B properties of the Laughlin state.
tionvalues ofenergyandangularmomentum arealmost
The authors thank Y. Kato,A.H. MacDonald,and A.
saturated.The energy gapfor the interactionII is found
Muramatsuforusefuldiscussion.S.S.acknowledgessup-
to be ∆E 0.09(e2/l ) by the exact diagonalization
≃ B portbyResearchFellowshipforyoungscientistsofJSPS.
method.Theinversetemperatureneededforconvergence
ThepresentworkissupportedbyGrant-in-AidforScien-
is so high (β & 6/∆E). This seems to be due to a high
tific Research (Grant No.1406899 and No.14740181) by
density of states for E &∆E.
the Ministry of Education, Culture, Sports, Science and
Technology of Japan.
]
el2 /B ∆exβa c=t 0.25 βT == 604
[ 1) D.C.Tsui,H.L.Stormer,andA.C.Gossard,Phys.Rev.Lett.
cle ∆β = 0.125 r) (∆ββ == 08.25) 48(1982) 1559.
rti (a)C(3 2) W.Pan,J.-S.Xia,V.Shvarts,D.E.Adams,H.L.Stormer,D.C.
a (b) Tsui,L.N.Pfeiffer,K.W.Baldwin,andK.W.West,Phys.Rev.
p
r Lett. 83(1999)3530.
e
y p 3) The Quantum Hall Effect second edition, edited by R.E.
g PrangeandS.M.Girvin(Springer,1990).
Ener 30 β4 0= kB5T0 [16/(0e2 /lB7)0] 0 2 4 6rarc 8 [lB1]0 12 45)) RJ..KB..JLaaiung,hPlhiny,s.PRhyesv..RLeevtt..L6e3tt(.159089()1918939). 1395.
6) G.MooreandN.Read,Nucl.Phys.B360(1991) 362.
7) B.I.Halperin,P.A.LeeandN.Read,Phys.Rev.B47(1993)
Fig. 3. Expectation values of (a) energy per particle and (b)
7312.
ODLRO correlation function C3(r). Results by the exact diag- 8) E.H.RezayiandN.Read,Phys.Rev.Lett.73(1994)900.
onalizationandfinite-temperature QMCcalculations areshown
9) A.A.Koulakov,M.M.Fogler,andB.I.Shklovskii,Phys.Rev.
for the QH system with 2s = 21 and Ne = 8 (ν = 1/3). rarc Lett. 76(1996)499.
denotes the arc-distance between r and the north-pole on the 10) D.Yoshioka,B.I.Halperin,andP.A.Lee,Phys.Rev.Lett.50
Haldanesphericalsurface.
(1983) 1395.
11) N.ShibataandD.Yoshioka,Phys.Rev.Lett.86(2001)5755.
12) F.D.M.Haldane,Phys.Rev.Lett. 51(1983)605.
Figure 3(a) shows the β-dependence of energy expec- 13) G.Fano,F.Ortolani,andE.Colombo,Phys.Rev.B34(1986)
tation value per particle in the canonical ensemble. The 2670.
14) E.Y.LohJr.andJ.E.Gubernates,inElectronicPhaseTransi-
results by the exact diagonalization and QMC calcula-
tions,editedbyW.HankeandYu.V.Kopaev(ElsevierScience
tions are in good agreement. For β & 60/(e2/l ), the
B PublishersB.V.,NewYork,1992).
expectationvaluesofenergyperparticlearealmostsatu- 15) S.E.Koonin,D.J.Dean,K.Langanke,Phys.Rep.278(1997)
ratedaswellasinFig.2(a).Sincetheexistenceoftheoff- 1.
diagonal long-range order (ODLRO) is one of the most 16) G.H. Lang, C.W. Johnson, S.E.Koonin, and W.E. Ormand,
remarkable properties of the Laughlin state,17 we calcu- Phys.Rev.C48(1993)1518.
17) S.M.GirvinandA.H.MacDonald,Phys.Rev.Lett.58(1987)
lated the ODLRO correlation function in the ν = 1/3
1252.
QH system with interaction II. The correlationfunction 18) E.H.RezayiandF.D.M.Haldane,Phys.Rev.Lett.61(1988)
is defined18 by C3(r′ r)= A†(r′)A(r) , where A(r) is 1985.
− h i