Table Of ContentComposite Pairings in Chirally Stabilized Critical Fluids
P. Azaria1 and P. Lecheminant2
1 Laboratoire de Physique Th´eorique des Liquides, Universit´e Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris, France
2 Laboratoire de Physique Th´eorique et Mod´elisation, Universit´e de Cergy-Pontoise, 5 Mail Gay-Lussac, Neuville sur Oise,
95301 Cergy-Pontoise Cedex, France
(Received: )
Westudyaone-dimensionalelectrongasinaspecialantiferromagneticenvironmentmadebytwo
0
0 spin-1/2 Heisenberg chains and the one-dimensional two-channel Kondo-Heisenberg lattice away
0 fromhalf-filling. Thesemodelsflowtoanintermediatefixedpointwhichbelongstotheuniversality
2 classofchirally stabilized liquids. UsingaToulouse pointapproach,theuniversalpropertiesofthe
modelsaredeterminedaswellastheidentificationoftheleadinginstabilities. Itisshownthatthese
n
models exhibit a non-Fermiliquid behavior with strong enhanced composite pairing correlations.
a
J PACS No: 75.10.Jm, 75.40.Gb
7
]
The concept of odd-frequency superconducting order- witha two-foldorbitaldegeneracyinteractingwitha pe-
l
e ingwasconceivedbyBerezinskii[1]whoconsideredspin- riodic array of localized spins i.e. the two-channel KHL.
-
r triplet odd-frequency pairing as an alternate of the con- It has been shown in Ref. [13] that these two models
st ventional theory 3He pairing. The possibility of such away from half filling exhibit a critical phase governed
. unusual pairing was revived nearly one decade ago by by a fixed point which belongs to the class of chirally
t
a Balatsky and Abrahams [2] in the context of singlet su- stabilized fluids [14]. In this work, we shall compute
m
perconductors and by Emery and Kivelson [3] in the correlation functions of physical observables by means
- two-channel Kondo problem with the emphasis laid on of a Toulouse point solution and characterize the dom-
d
the composite nature of the order parameter for odd- inant instabilities of both models which turn out to be
n
o frequency pairing. This idea of composite pairing was of a composite nature. Very recently, the one dimen-
c further developped by Coleman et al. [4] who proposed sional N-channel KHL has been investigated by Andrei
[ that heavy-fermion superconductors might involve odd- and Orignac [15] using a conformal field theory (CFT)
frequency triplet pairing. approach. These authors found that the leading insta-
1
v One of the most important difficulty in finding some bility is of a composite pairing type when N 4 [15].
≤
2 specific lattice models for realization of odd-frequency In this respect, the exact solution at the Toulouse point
7 pairing stems from the fact that it requires a controlled provides an independent and physically transparent ap-
0 analysis in the strong coupling regime. Such an ap- proach for the particular 2-channel KHL.
1 proach is possible in some extreme limits as in one di- The models. The first model (referredin the following
0
mensionwherenon-perturbativetechniquesareavailable. as a 1DEG in a special active environment) consists of
0
Strong odd-frequency singlet pair correlations were, in a 1DEG away from half filling coupled symmetrically by
0
/ particular, identified within the bosonization approach a Kondo coupling (JK > 0) with two non-interacting
t
a [5] of the one-dimensional single channel Kondo lattice antiferromagnetic (JH >0) spin-1/2 Heisenberg chains:
m in an anisotropic limit [6,7]. Another candidate is the
2
onedimensionalsinglechannelKondo-Heisenberglattice
d- (KHL) which consists of a one dimensional electron gas H1 =−tX(cid:16)c†iσci+1σ+H.c.(cid:17)+JKXXSci·Sai
n (1DEG) interacting with an antiferromagnetic Heisen- i a=1 i
o berg spin-1/2 chain by a Kondo coupling. Away from 2
:c half filling, this model has a spin gap [8] and is thus ex- +JHXXSai·Sai+1. (1)
v pected to exhibit enhanced pairing correlations. In par- a=1 i
i
X ticular, ithas been shownrecently [9]that the dominant Here,c istheconductionelectronoperatoratsiteiwith
iσ
ar ionfstthaebipliatireasmaerteeorsf.aIcnotmhpisosleittetenra,twuereshinalal csteurtdayintwreogidmife- spinσ =↑,↓,S~ci =c†iα~σαβciβ/2(~σ beingthePaulimatri-
ces)standsfortheelectronspinoperatoratsiteiwhereas
ferent generalization of the one dimensional KHL with
the localizedspinoperatoratsite i onthe chainofindex
two channels which have no spin gap but exhibit domi- a = 1,2 is denoted by S . In the limit J t,J ,
nant unconventional pairing instabilities. A first gener- ai K ≪ H
the low energy behavior of the model can be determined
alizationistoconsidera1DEGcoupledbyaKondocou-
using the continuum limit of the electron operator in
pling to two non-interacting antiferromagnetic Heisen-
terms of right and left moving fermion fields: c
berg spin-1/2 chains. This model belongs to the more iσ →
general class of Luttinger liquids in active environments RσeikFx+Lσe−ikFx,x=ia0(kF beingtheFermimomen-
tumanda0 the latticespacing). The continuumdescrip-
introduced in the context of the striped physics [10–12]. tion of the conductionspin density operatorS can then
A second generalization consists of conduction electrons beexpressedintermsofaspincurrentJ0R,Lbeclongingto
1
the SU(2)1 Kac-Moody (KM) algebra [5] and a bosonic IR,L = −i ξ~R,L ∧ ξ~R,L/2,J1R,L − J2R,L = iξ~R,LξR0,L.
fieldΦc =ΦcL+ΦcR thataccountsforthechargedegrees The Hamiltonians (2,4) share then the same structure:
obfeifnregetdhoems:taSgcg→ereJd0mRa+gnJe0tLiz+atcioosn(.2kInFxth+e√sa2mπeΦwc)any0,,tnh0e Hto1n,i2an=oHf c0th+eHch0a(ξrg0)e+boH¯sonwihcefireeldHsc0ainsdthe0f(rξe0e)Hisamthile-
spin densities of the surface chains are represented as free Hamiltonian of the massless MajoranHa fermion ξ0.
(see Refs. [16,5]): Sa(x)=Ja(x)+( 1)x/a0na(x) where All nontrivial physics is incorporated in ¯ which sepa-
Ja =JaR+JaL and na are respectiv−elythe uniform and ratesintotwocommutingandchirallyasyHmmetricparts:
staggered parts of the magnetization. The chiral spin ¯ = ¯1+ ¯2, [¯1, ¯2]=0 with
cWsiutriytrheonftthstihsJealoRlaw,tLteicbneeerlmognyogddeaelslsc(or1i)ptotfiootrnh,ientcShoUem(H2ma)e1mnKislutMorantiaaelngfieldbliernnag-. H H H¯1H= π2vH1 I2RH+ 2π3v0 J20L+gJ0L·IR (5)
reads as follows in the continuum limit: with v1 = vH,v0 = vF for the 1DEG in a special active
v 2πv environment and the reverse for the two-channel KHL;
H1 ≃ 2F (cid:16)(∂xΦc)2+(∂xΘc)2(cid:17)+ 3F (cid:0)J20R+J20L(cid:1) ¯2 is obtained from ¯1 by inverting chiralities of all
H H
the currents. For an antiferromagnetic Kondo coupling
2
+ 2π3vH X(cid:0)J2aR+J2aL(cid:1)+g(J0R·IL+J0L·IR) (2) v(gan>t. 0N)a,ivtheleyionnteerawcitlilntghupsaretxpoefcHt¯thisatmtahregimnaoldlyelrweliel-l
a=1
enter a massive strong coupling region. However, due
where we have neglected all oscillatory, marginal irrele- to the chiral asymmetry of ¯, the effective interaction
vantcontributionsaswellascurrent-currentinteractions H
flows towards a conformally invariant intermediate fixed
ofthesamechirality. AsshowninRef.[13],atthestrong
point with a smaller central charge than in the ultravi-
couplingfixedpoint,theselatterinteractionsonlyleadto
olet. Stated differently, the interaction produces a mass
renormalizationofvelocitiesandlogarithmiccorrections.
gap in some sectors but there are still some degrees of
In Eq. (2), Θ = Φ Φ is the dual charge field, v
c cL− cR F freedom that remain critical in the infrared (IR). The
the Fermi velocity, v the spin velocity of the magnetic
H nature of the fixed point of chirally asymmetric current-
epnlivnigrocnomnesntatn,ta;nIdRg,L≃=JJK1aR0,L>+0Ji2sRt,hLeisinaterSaUct(i2n)g2 cKoMu- cmuirnreedntbymAodnedlrseiweitthalt.h[1e4s]tbryucatucroem(b5i)nahtaiovne obfeCenFTdeatenrd-
current being the sum of two SU(2)1 KM currents. Bethe ansatz techniques. This fixed point belongsto the
The second model considered in this letter is the one-
class of chirally stabilized fluids introduced in Ref. [14]
dimensional two-channel KHL:
describing the IR behavior of one-dimensional interact-
2 ingchiralfermions. Asimplerwaytoidentifythecritical
H2 =−tXX(cid:16)c†iaσci+1aσ +H.c.(cid:17)+JHXS0i·S0i+1 degrees of freedom of model (5) in the IR limit can be
i a=1 i done by means of a Toulouse point approach as in the
2 two-channel Kondo model [3]. By allowing anisotropic
+JKXa=1Xi Saci·S0i (3) itnotneiraanct¯iocna(ngb→emgak,pgp⊥ed),otnhteoUfr(e1e)MvearjsoiroannaoffetrhmeiHonasmfiolr-
H
wherenowtheelectronoperatorciaσ hasachannelindex a special value of the interaction g∗. Although the posi-
a = 1,2 and interacts by a Kondo coupling (J > 0) tion of the solvable point is non-unkiversal, the Toulouse
K
with a periodic array of localized spins S0i. Away from limitsolutioncapturesthephysicalanduniversalproper-
half filling, the continuum limit of the Hamiltonian (3) ties of the models including the SU(2) case. The details
proceeds in the same way as in the first model and with oftheapproachcanbefoundinRefs.[18,13]andweshall
the same degree of approximations than in Eq. (2), one now briefly review it to fix the notations. The starting
has: point of the Toulouse solution is the Abelian bosoniza-
v 2 2πv tion of all SU(2)1 currents Ja,a = 0,1,2 in terms of
H2 ≃ 2F X(cid:16)(∂xΦac)2+(∂xΘac)2(cid:17)+ 3H (cid:0)J20R+J20L(cid:1) masslessbosonic fields ϕa [5]: JazR,L =(1/√2π)∂xϕaR,L,
a=1 Ja+R,L = (1/2πa0)e∓i√8πϕaR,L. Introducing the sym-
2
+ 2π3vF Xa=1(cid:0)J2aR+J2aL(cid:1)+g(J0R·IL+J0L·IR) (4) mtheetrSicU(c2o)m2bcinuarrteionnt oIfwthrietefis:eldIsRz:,Lϕ+==(1(/ϕ√1π+)∂ϕxϕ2)+/R√,L2,,
where Φ is a bosonic field (Θ being the dual field) IR+,L = (i/√πa0)ξR3,Lκe∓i√4πϕ+R,L. A fermionic zero-
ac ac mode operator κ has been introduced to ensure the
associated with charge fluctuations in the ath channel.
proper anticommutation relations. The following canon-
ThechiralSU(2)1currentsJaR,Lcorrespondtothechiral ical transformation is then performed:
uniform part of the electron spin density whereas J0R,L
are the chiral SU(2)1 currents of the local moments. ϕ0 =√2Φ¯2 Φ¯1, ϕ+ =√2Φ¯1 Φ¯2
Toulouse point solution. The next step of the ap- − −
proach is to use the representation of two SU(2)1 cur- ϑ0 =√2Θ¯2+Θ¯1, ϑ+ =√2Θ¯1+Θ¯2 (6)
rents interms offour Majoranafermionsξ0 andξ~[5,17]: where ϑ0 and ϑ+ (respectively Θ¯1 and Θ¯2) are the dual
2
fields associated with ϕ0 and ϕ+ (respectively Φ¯1 and correlationfunctionshavebeencomputedinRef.[13]and
Φ¯2). For a special positive value (Toulouse point) of the slowest ones are the staggered parts that decay as
g , (g∗ = 4π(v0 + v1)/3), the arguments of the inter- x−3/2. Using the previous Abelian bosonization of the
akctingkterms become those of free fermions so that they chiral fermions and the Toulouse basis (6), we find in
can be refermionized further with the introduction of a the electronic sector the following representation for the
pair of Majorana fields (η,ζ) using the correspondence: charge density wave (CDW), spin density wave (SDW),
ηR,L +iζR,L = (κ/√πa0) e±i√4πΦ¯2R,L. One finally ends singlet (SS) and triplet (ST) superconducting order pa-
rametersin terms ofthe different criticalfields at the IR
with:
fixed point:
H¯ −= iu2v211(cid:2)(cid:2)(ξ∂R3x∂Φ¯x1ξ)R32−+ξ(∂L3x∂Θx¯ξ1L3)2(cid:3)(cid:3)−−iui2u222[η[ζRR∂∂xxηζRR−−ηζLL∂∂xxηζLL]] OCDW =OL~†σSRDWσ ≃=eLi√†α2~σπαΦβcRcoβs≃(cid:16)√ie2iπ√Φ2¯π1Φ(cid:17)cζζLRζζRL
+im(cid:2)ξR3ηL−ηRξL3(cid:3) (7) hcos(cid:16)√2πΘ¯1(cid:17),sin(cid:16)√2πΘ¯1(cid:17),sin(cid:16)√2πΦ¯1(cid:17)i
w(2hve0re mv1)/=3.g⊥T/h2eπTao0,uluou1se=so(l2uvt1io−n ivs0)s/ta3b,leanpdrouv2ide=d OSS =−iLα(σy)αβRβ ≃e−i√2πΘccos(cid:16)√2πΦ¯1(cid:17)ζRζL
tThhaet−fiarlsltvtewloocitteiersmsarienpEoqs.itiv(7e)id.ee.scr1i/b2e d≤ecvo0u/pvl1ed≤fre2e. O~TS =−iLα(~σσy)αβRβ ≃ie−i√2πΘcζRζL
massless bosonic and Majorana fields, Φ¯1 and ζ, con- hcos(cid:16)√2πΘ¯1(cid:17), sin(cid:16)√2πΘ¯1(cid:17),sin(cid:16)√2πΦ¯1(cid:17)i. (9)
−
tributing to criticality with the central charge: c = 3/2.
The remaining part of the Hamiltonian (7) has a spec- All these conventional order parameters have thus the
tral gap m and describes hybridization of the Majorana same scaling dimension (2) and the corresponding pair
ξ3 and η fermions with different chiralities. Adding the correlation functions decay as x 4 i.e. much faster
−
contribution of the excitations that do not participate than in the one dimensional metals. One should notice
in the interaction, the total central charge in the IR of that the Z2 pseudocharge excitations contribute in (9)
the 1DEG in a special active environment (respectively through the density energy operator iζ ζ . In a critical
R L
the two-channel KHL) is c = 3 (respectively c = 4). Isingmodel,thereareotherprimaryfields(orderanddis-
Apart from the charge degrees of freedom, the elemen- order parameters) that have a smaller scaling dimension
tary excitations of the models in the IR are of different (1/8) which are highly nonlocal in terms of the original
nature: The magnetic excitations are spinons defined as Majorana fermion [5]. This leads us to investigate the
π/2kinksofthebosonicfieldΦ¯1describinganeffective possibility that the ground state might be characterized
p
S=1/2Heisenbergchainandnonmagneticsingletexcita- bycompositeorderparameters: staggeredodd-frequency
tionsassociatedwiththetwomasslessMajoranafermions singletpairing (c-SP)andstaggeredcomposite CDW (c-
ξ0,ζ (critical Ising degrees of freedom) referred as pseu- CDW). At the IR fixed point, we find the following cor-
docharge degrees of freedom in Refs. [18,13]. These Z2 respondence for these order parameters:
excitations play a crucial role in the nontrivialIR physi-
cal properties of the models as we shall see now. c SP = ~TS (n1+n2) e−i√2πΘcµ0µ4
virPohnymseicnatl.pTrohpeeGrtrieeseno’fstfhuen1cDtioEnGfoinr athsepericgiahlt-alcetftivmeoenv-- OcO−C−DW =OO~SDW· ·(n1+n2∼)∼ei√2πΦcµ0µ4 (10)
ingfermionscanbecomputedusingthebosonicrepresen-
tation: (R,L)σ = (κσ/√2πa0) eiτσπ/4e±i√4πϕσR,L where w(rhesepreecµti0v,eµly4 o(rredsepre)cptiavrealmyeσt0e,rσs4a)ssaorceiattheedIwsiintgh dthiseorMdaer-
τ , = 1, ϕσR,L = (ΦcR,L +τσϕ0R,L)/√2, and κσ are jorana fermions ξ0,ζ. ¿From Eq. (10), we deduce the
K↑le↓in-fa±ctorstoinsurethecorrectanticommutationrela- leading asymptotics of the correlation function corre-
tions between fermion fields of different spin index [19]. sponding to the composite order parameters:
Using the Toulouse basis (6), one then obtains at the IR
fixed point the estimate: hOcomposite(x,τ)Oc†omposite(0,0)i∼
1
hRσ(x,τ)Rσ†′ (0,0)i∼ (v2τ2+x2)1/2(v2τ2+x2)1/8(u2τ2+x2)1/8 (11)
δ ′ F 1 2
σ,σ
(8)
(vFτ ix)1/2(u1τ +ix)1/2(u2τ ix) sothatthecompositeorderparametersinducethedomi-
− − nant instabilities and have enhanced long-range coher-
which means that the system displays non-Fermi liquid ence. These fluctuations can be made even more en-
properties; the left-moving Green’s function is obtained hanced upon switching on a very small ferromagnetic
from Eq. (8) by the substitution: i i. The most in- (g12 < 0) exchange interaction between the spin chains:
terestingphysicalquantitiesarethec→orr−elationfunctions 12 =g12S1 S2. Indeed, at the IR fixed point, this per-
O ·
of the various possible order parameters. The spin-spin turbation mostly affects the pseudocharge sector leaving
3
finertmacitontheξ0maacgqnueitriecsoanep:oOsi1t2iv≃e gm1a2isξsR0iξ.Le0.. TithseaMssaojcoiraatneda pinasrtaambielitteiress iosfstthielltgwivoe-cnhabnyneElqK. H(L11a)r.e tThhues odfomaicnoamnt-
Ising model is in the disorder phase: µ0 = 0 and the positenatureasinthe 1DEGinaspecialactiveenviron-
h i 6
Z2 symmetry with respect to the interchange of the two ment. Finally, one should note that the scaling dimen-
chainsisnowbroken. Allotherdegreesoffreedomremain sionsofthephysicalobservablesofthetwo-channelKHL
criticalsothatthepaircorrelationoftheconventionalor- obtained by the Toulouse point solution coincide with
der parameters (9) still decay as x 4 whereas from Eq. those of the recent CFT approach [15]. In this respect,
−
(10) one immediately observes that the composite ones we stress that, in contrast with the latter approach, the
decay now as x−5/4 instead of x−3/2 at g12 =0. Toulouse point solution unables us to take into account
The two-channelKHL case. The determination of the the inherent velocities anisotropy in the problem.
ground-statephysicalpropertiesofthetwo-channelKHL In summary, we have shown that the two generaliza-
proceeds in the same way as in the previous model. We tions of the one dimensional KHL considered in this let-
firstusetheAbelianbosonisationoftheright-leftmoving terexhibitanontrivialnon-Fermiliquidlow-temperature
fermions: (R,L)aσ = (κaσ/√2πa0) eiτσπ/4e±i√4πϕaσR,L phase belonging to the class of chirally stabilized fluids
where ϕ = (Φ + τ ϕ )/√2, and κ are withstrongenhancedstaggeredcomposite pairingcorre-
aσR,L acR,L σ aR,L aσ
some Klein-factors [19]. The Green’s functions of the lations.
chiral fermions have a similar structure than in Eq.
(8) apart from they fall now with the distance with
a power 5/4. The leading asymptotics of the corre-
lation function of the localized spins have been com-
puted in Ref. [13] and in particular the staggered part
decays as x 3. Using the Toulouse basis (6) and in-
− [1] V. L. Berezinskii, JETP Lett. 20, 287 (1974).
troducing the charge and relative charge bosonic fields [2] A. Balatsky and E. Abrahams, Phys. Rev. B 45, 13 125
Φfo±rctRh,eLel=ect(rΦo1nciRc,dLe±greΦe2scoRf,Lfr)e/e√do2m, tchaenoarldsoerbpeaerxapmreestseerds [3] V(1.9J9.2)E;mEe.rAybarnadhSam.As.eKtiavle.,lsiobnid,P5h2y,s1.2R71ev(.1B99456).,10812
in terms of the critical fields at the IR fixed point. We
(1992).
findthefollowingcorrespondencefortheconventionalor-
[4] P. Coleman, E. Miranda, and A.M. Tsvelik, Phys. Rev.
der parameters: Lett. 70, 2960 (1993); Phys. Rev. B 49, 8955 (1994);
Phys. Rev.Lett. 74, 1653 (1995).
OCDW =L†aσRaσ ∼ei√πΦ+ccos(cid:16)√2πΦ¯1(cid:17)IΦ [5] Forarecentreview,see: A.O.Gogolin,A.A.Nersesyan,
andA.M.Tsvelik,BosonizationandStronglyCorrelated
O~SDW =L†aα~σαβRaβ ∼ei√πΦ+c Systems, Cambridge UniversityPress (1998).
[6] O. Zachar, S. A. Kivelson, and V. J. Emery, Phys. Rev.
hcos(cid:16)√2πΘ¯1(cid:17)IΦ†,sin(cid:16)√2πΘ¯1(cid:17)IΦ†,−sin(cid:16)√2πΦ¯1(cid:17)IΦi Lett. 77, 1342 (1996).
OSS =−iLaα(σy)αβRaβ ∼e−i√πΘ+ccos(cid:16)√2πΦ¯1(cid:17)IΘ† [7] PC.onCdo.leMmaatnt.,9A,.34G5eo(1rg9e9s7,).and A. M. Tsvelik, J. Phys.
O~TS =−iLaα(~σσy)αβRaβ ∼e−i√πΘ+c [8] SA..RE..WSihkikteemanad, II..AAfflffleecckk,,Panhdys.SR. Rev..WBh5i4te,,98P6h2y(s1.9R9e6v);.
h−cos(cid:16)√2πΘ¯1(cid:17)IΘ,sin(cid:16)√2πΘ¯1(cid:17)IΘ,sin(cid:16)√2πΦ¯1(cid:17)IΘ†i [9] OLe.tZt.a7ch9a,r9a2n9d(1A9.97M).. Tsvelik, cond-mat/9909296.
[10] V. J. Emery, S. A. Kivelson, and O. Zachar, Phys. Rev.
fiwIΘehldeirseΘoIbΦct.a=inTeµhd0eµf4rsoccamolsin(I√gΦπdΦbimy−ecrn)es+piolaniσci0onσfg4asΦliln−t(ch√ebπsyeΦi−ctocs)ndvauennad-l [11] APBh.5yH6s.,.R6C1ea2vs0.trL(o1e9tNt9.7e)7t;o9,i,bC4id6.25d99e(,1C195.97C6)4h.1am(1o9n9,9)a.nd C. Nayak,
tional o−rder parameters is 5/4 and their corresponding [12] M. Granath and H. Johannesson, Phys. Rev. Lett. 83,
pair correlation functions decay thus as x 5/2 i.e. have
− 199 (1999).
noenhancedlong-rangecoherence. Notice,however,that [13] P. Azaria and P. Lecheminant, cond-mat/9912406.
this decay is much slower than in the 1DEG in a special [14] N. Andrei, M. R. Douglas, and A. Jerez, Phys. Rev. B
active environment case. Similarly, we find the follow- 58, 7619 (1998).
ing description at the IR fixed point for the staggered [15] N. Andreiand E. Orignac, cond-mat/9912372.
composite order parameters: [16] I. Affleck,Nucl. Phys.B265, 409 (1986).
[17] D. Allen and D. S´en´echal, Phys.Rev.B 55, 299 (1997).
c SP = ~TS n0 e−i√πΘ+c Θ [18] P. Azaria, P. Lecheminant, and A. A. Nersesyan, Phys.
OOc−−CDW =OO~S·DW ∼·n0 ∼ei√πΦI+c [19] TRheve.pBro5d8u,ctRκ8↑8κ81↓ h(1a9s9n8o).dynamicandwefixitequalto
µ0µ4cos √πΦ c +3iσ0σ4sin √πΦ c (12) itorecoverthebosonicexpression oftheSU(2)1 current
(cid:0) (cid:0) − (cid:1) (cid:0) − (cid:1)(cid:1) J0R,L used in this work; similarly, in the two-channel
so that the scaling dimension of these operators is 3/4 KHL, theprescription κa↑κa↓ =i (a=1,2) is used.
and the pair correlation function of the composite order
4
