Table Of ContentKlein Factors in multiple Fractional Quantum Hall edge tunneling
R. Guyon1, P. Devillard1,2, T. Martin1,3 and I. Safi1
1 Centre de Physique Th´eorique, Case 907 Luminy, 13288 Marseille Cedex 9, France
2 Universit´e de Provence, 13331 Marseille Cedex 03, France
3 Universit´e de la M´editerran´ee, 13288 Marseille Cedex 9, France
2
0 A fractional quantum Hall liquid with multiple edges is considered. The computation of trans-
0 port quantities such as current, noise and noise cross correlations in such multiple edge samples
2
requires the implementation of so called Klein factors, which insure the correct quasiparticle ex-
n change properties. The commutation relations of these factors are obtained by closing the system
a into a single edge. The non-equilibrium Green’s function formalism associated with such factors is
J derived for a simple Laughlin fraction of the Hall effect. It is shown explicitly how Klein factors
4 enterthecalculation ofthenoisecrosscorrelations,aswellasthecorrectiontothePoisson limitfor
1 the noise.
] The edge state picture of the fractional quantum Hall
l
al effect [1] (FQHE) has led to fundamental discoveries. Φ1 Φ2 Φ3 ΦN
h Various systems have been studied to analyze the quasi-
- particle excitations which tunnel through such quantum s’1 s’2 s’3 s’N
s
e fluids [2,3]. In particular, for a point contact geometry
m with twoedges,the detectionofthe backscatteringnoise T T T T
1 2 3 N
. leads to the measurementof the quasiparticlecharge [4].
t
a Atthe same time, resonantantidottunneling geometries
m
have allowed a capacitive measurement of the charge [5] s1 s2 s3 sN
- in multiple edge geometries. Aside from this situation,
d
multi edge geometries have not receivedmuch attention.
n
o Yet it was argued recently [6] that the detection of noise
c correlations in a three edge system constitutes a direct Φ
[ link to the quasiparticles statistics. One edge can be de- 0
2 pleted due to the tunneling of quasiparticles into other FIG. 1. Multiple edge geometry: tunneling processes Ti
v edges. TheHanbury-BrownandTwisstypegeometry[7] between the injecting edge specified by φ0 and other edges,
0 of Ref. [6] probes whether two quasi-holes on the inject- with fields labelled φ1, φ2,..., φN. Curvilinear abscissas
4 ing edge are allowed to overlap. Here, the key point is (s,s′1,s′2,...) are attached to each edge. In our disconnected
3 that quasiparticle edge tunneling operators need to be edge geometry only thefull lines definingtheedges are kept.
9
Whenconsideringtheconnected,singleedgeproblemthedot-
0 supplemented by Klein factors in order to specify the
ted lines are added to fix the topological constraints. Inside
1 statistics. Klein factors were so far mostly introduced
0 to tackle fermionic problems [8], yet it seems that their frame shows how the transport properties in T2 are affected
/ bythe fluctuatingcurrent in T1.
t implementation is quite relevant for complex fractional
a
Hall geometries.
m Our starting point is the Hamiltonian [9] which
The purpose here is to give a pedagogicalview of how describes a system of decoupled edges H =
- 0
d fractionalstatisticsentersthesemultipleedgegeometries: (v ¯h/4π) N ds(∂ φ )2 with s the curvilinear ab-
n F l=0 s l
o which (fractional) phase is generated when two quasi- scissa. ThPe fieldRs ∂sφl enter the description of long wave
particles from different edges are exchanged ? We fo- length excitations along each edge.
c
: cus on Laughlin filling factors ν = (2p+1)−1 (p inte- Tunneling between edge 0 and the other edges is de-
v
ger). First,specifyingaquasiparticletunneling Hamilto- scribed by the hopping Hamiltonian [3]:
i
X nian,wederivethealgebrawhichisobeyedbyKleinfac-
r tors. Next, we introduce the Keldysh Green’s functions N
a for the Klein factors. These tools are then applied to Ht = Γle−iǫe∗χl/h¯cTl(ǫ) (1)
study the non-equilibrium currentandnoise correlations Xl=1ǫX=
±
in a multi-edge system. In particular, it will be shown
that Klein factors are unnecessary for simple, two edge where Tl = Tl(+) = ψ0†(sl)ψl(s′l) = Tl(−)†. ψl† cre-
samples. Finally, we argue that when considering the ates a quasiparticle excitation with charge e = νe on
∗
non-equilibriumaverageofa tunneling operatorbetween edge l. Here “0” refers to a specific edge where, for
twoedges,thereremainsalwaysacontributionassociated instance, current can be injected into the others. A
withKleinfactors,whichissymptomaticofquasiparticle voltage bias can be imposed as usual using the Peierls
exchange with other edge “reservoirs”. substitution, i.e. e χ (t) = h¯ω ct for a constant DC
∗ l 0
1
bias V = h¯ω /e imposed between edges 0 and l. The tied to the edge configuration. Using Eq. (4), we find a
l 0 ∗
quasiparticle operators are written in a bosonized form connection between the elements p and the sign of the
jk
ψl(s) = (2πα)−1/2e−i√νφl(s) with α a short distance algebraicdistance : sgn(sk−sj)=pk0+p0j+pjk. If for
cutoff. Note that no fractional statistics is taken into instance k =2 and j = 1 with sgn(s −s )= 1, we can
2 1
account when quasiparticles from different edges are ex- choose : p =−1=−p =−p .
01 02 12
changed, whereas [10]: These statistical constraints are now applied to a de-
fine problem of fractional Hall edge transport: the com-
ψl(s)ψl(s′)=e±i√νπsgn(s−s′)ψl(s′)ψl(s) (2) putationofcurrentandnoisecorrelationsassociatedwith
severaltunneling processes. We returnto N+1edges in
on a given edge. Fractionalstatistics is a consequence of
the previoussystemwith N tunnelings paths. We repre-
the Laughlin correlatedstate describing the FQHE. The
sent the current on lead l using the Keldysh formulation
commutationrelations for the ψ ’s should originate from
l
of non-equilibrium transport:
the derivation Chern–Simons boundary action. Here,
wilaerctoonjtehcotsuerefotrhaatstyhsetemcomofmcuotnantieocnterdeleadtgioens.s aFroerstihmis- hIli= 1 hTKIl(tη)e−h¯i KHt(t′)dt′i (5)
2 R
reason, Klein factors are introduced, which means that ηX=
±
each quasiparticle operator ψ is changed to F ψ . The
l l l
where η labels the branch of the time ordering contour
requirement of unitarity, Fl†Fl =FlFl† =1, is consistent K.
with the fermionic case (ν =1).
The cross current fluctuations (i.e. the noise correla-
Consider a closed system (Fig.1) where edges are con-
tions) between two leads j and k are given by:
nected with an open contour which brings all the edges
into one. Klein factors are not necessary in this system
because the commutation relations of tunneling opera- Sjk(t,t′)= hTKIj(tη)Ik(t′−η)e−h¯i KHt(t1)dt1i (6)
R
torsareenforcedbythesinglechiralbosonicfieldφ. Yet, ηX=
±
bylookingatthe commutationrelationsofdifferenttun-
The perturbative computation of these operators is
neling operators we show below that for a system with
achieved by expanding the evolution operator. The fi-
thesametopologybutwhereedgesaredisconnected,the
nite frequency noise S (ω) = S(n)(ω) (nth order in
Klein factors can be computed. The operator describing jk n jk
tunneling from edge “0” (location s ) to edge “l” (loca- Γj’s) is the Fourier transform oPf the real time correlator
l
tion s′l) reads Tl =(2πα)−1ei√νφ(sl) e−i√νφ(s′l) . (Eq. (6)). The (n+2)th order of the noise is :
Using the Baker–Campbell–Hausdorff formula [11],
one can compute the commutator of two tunnel oper- S(n+2)(ω)= dteiωt Γ(jǫj)Γ(kǫk) − i n dt..dt
ators: jk Z n n! (cid:16) ¯h(cid:17) Z 1 n
[Tj,Tk]=e−iπνTkTj 1−e−iπν[sgn(sj−sk)+sgn(s′j−s′k)] × ǫǫ′eiǫω0t−iω0 nl=1ǫltl
h i ηηX1..ηnXǫǫ′ ǫ1X...ǫn P
(3) N
Chirality imposes that sgn(s −s ) = −sgn(s −s ) if ×hTKTj(ǫ)(tη)Tk(ǫ′)(0−η) Γ(qǫ11)Tq(1ǫ1)(tη11)....
k j ′j ′k qX1=1
the twotunneling pathsdonotcross. Inthis convention,
N
whichwillbekeptthroughoutthepaper,[Tj,Tk]=0. In ... Γ(ǫn)T(ǫn)(tηn)i−hI (t)ihI (0)i (7)
tchroesosipnpgo,sTitjeTksit=uaet2iioπnνswgnh(sejr−estkh)TektTujn.neling paths have a qXn=1 qn qn n j k o
We now turn to the same system (same topology) de- The lowest non-vanishing contribution to the noise in a
picted in Fig. 1 but where all edges aredisconnected. In given terminal (j =k) is of order 2 in the tunneling am-
order to enforce fractional statistics, we introduce Klein plitude, i.e. the Poisson result. The cross correlations
factorsinsuchawaythatthetunnelingoperatorsbecome require to compute the 4th order contribution. For ex-
Tl =ψ0†(sl)ψl(s′l)F0Fl†. We requirethat the commutator ample,thenon–vanishingcontributiontonoisecorrelator
of the tunneling operators of this disconnected system between two terminals (j <k) :
give the same results as the connected one. The Klein
factors thus have to satisfy the relation : −2e 2 −i 2
S(4)(t,t)= ∗ ǫǫ|Γ Γ |2η η dt dt
(F0Fk†)(F0Fj†)=eiνπsgn(sk−sj)(F0Fj†)(F0Fk†) (4) jk ′ ¯h2 (cid:16) ¯h (cid:17) ηXη1η2ǫǫX′ǫ1ǫ2 ′ j k 1 2Z 1Z 2
hT T(ǫ)(tη)T(ǫ′)(t′ η)T(ǫ1)(tη1)T(ǫ2)(tη2)i
Eq. (4) is compatible with fractional statistics, i.e. K j k − j 1 k 2
FjFk = e−iπνpjkFkFj, for j 6= k where pjk = ±1. For
arbitrary j and k, pjk are the elements of an antisym- expiǫχj(t)+iǫ′χk(t′)+i ǫlχl(tl) (8)
metric N ×N matrix with elements 0,1,−1 which are lX=j,k
2
To this order, the Keldysh ordered product contains the presentisgη1η2(t)=π/2. Togetherwiththerequirement
11
productoftwocontributions. Oneofsuchoriginatesfrom of Eq. (12) to give a non-zero result, the term in the ex-
the computation of the average of the bosonic fields as- ponential then vanishes. This completes the proof that
sociated with each edge. This contains the dynamical for a two edge system, when computing non-equilibrium
aspectofthesefields,andgivesrisetoexponentiatedchi- averages for the current, the density fluctuations or the
ral Green functions. The other factor has no dynamics noise, the Klein factors do not contribute [12], and this,
other than that specified by time ordering; It contains order by order in the perturbation series.
the Klein factors: Wenowturntothecomputationofthe zerofrequency
hTK Tj(ǫ)(tη)Tk(ǫ′)(t′−η)Tj(ǫ1)(tη11)Tk(ǫ2)(tη22) i= c2uirnrepnrtescernocssecoofrareDlaCtiobnisasb.eEtwqe.e(n8t)uinsnineltienggraptaetdhsov1erantd:
n o
(2πα)−4hTKe−i√νǫφηj(0,t)e−i√νǫ1φηj1(0,t1)i 2e 2
××hhTT×KKeee+−+i√ii√√νννǫǫǫ1φ′φφη0η0−k(1−η(−(s0ks,jtk/′j)2/e,2t−,)tei1√+)eνi+√ǫ2iνφ√ǫηk′ν2φǫ(−020φη,tη0(2s2)k(ijs/k2j/,t2′,)t2)i S|˜(Γ2jkjπΓ(αωk)|4=2 e0i)ω0=[ǫ(t−¯ht4∗1)−ηXηǫ′1tη22]eXǫ2ǫν′[Gǫηǫη′η11(0η,2t−Zt1d)+tZG−dηtη12Z(0,d−tt22)]
×hTK(F0Fj†)(ǫ)(tη)(F0Fk†)(ǫ′)(t′−η) × eνǫǫ′[Gηη2(−skj,t−t2)+G−ηη1(skj,−t1)]eiξ−1 (13)
×(F0Fj†)(ǫ1)(tη11)(F0Fk†)(ǫ2)(tη22)i , (9) (cid:16)eνǫǫ′[Gη−η(−skj,t)+Gη1η2(−skj,t1−t2)] (cid:17)
where s =s −s . The contraction of the chiral fields Where the Klein factor time ordered product reads :
kj k j
gives rise to quasiparticle conservationlaws ǫ=−ǫ and
ǫ′ = −ǫk . It is most convenient to compute thejtime eiξ =hTKe−iǫ√νθj(tη)e−iǫ′√νθk(0−η)eiǫ√νθj(tη11)eiǫ′√νθk(tη22)i.
ordered Klein factor product using a bosonization for-
(14)
mulationsimilar to that used for the quasiparticlefields:
F0Fk† ≡e−i√νθk . (10) toNbeotseubthtraatcttheedpbreofodruecctoomfpcuutrirnegntthaeveFroaugreiserhItjriahnIskfiorhmas.
Introducing the fact that sgn(s −s ) = 1 in Eq. 4 , The resulting expression then identifies with the noise
k j
we find (using the Hausdorff formula) that [θ ,θ ] = iπ. correlator which is typically measured in experiments
j k
From the fields θ , annihilation and creation operatorsa [13]. Because of the symmetry properties of the Keldysh
i
and a can then be defined as a ≡ 1 (θ +iθ ), with Greens function, only the configuration η = −η = −η
† √2π j k 1 2
the usual property that ha†ai0 =0 (“ground” state) and needs to be considered and the configuration ǫ = −ǫ′ is
chosen. Indeed,thisturnsouttobetherelevantconfigu-
haa i=1.
†
The ground state expectation value of products of ration which generates a series solution of the noise cor-
the bosonic fields reads: hθ θ i = −hθ θ i = iπ and relations [6]. Choosing the Klein factor correlator with
j k 0 k j 0 2
hθ θ i=hθ θ i= π. ǫ=+:
j j k k 2
poBrtecsaituusaetiwoen,araeKdeealdliynsghwmitahtraixniosn-ienqtruoildiburcieudmfotrratnhse- hTKe−i√νθj(tη)ei√νθk(0−η)ei√νθj(tη11)e−i√νθk(tη22)i=
bosonic fields of Eq.(10) : =exp ν gjηk−η(t)+gjηj−η(t−t1)−gjηkη(t−t2)
h (cid:16) (cid:17)i
gjηk1η2(t)≡hTK θj(tη1)θk(0η2) i ×exp ν −gk−jη−η(−t1)+gk−kηη(−t2)+gj−kηη(t1−t2)
n o h (cid:16) (cid:17)i
π η1+η2 η1−η2 iνπ
=i sgn(t)− . (11) =exp η(sgn(−t )−sgn(t−t )) (15)
2n 2 2 o (cid:20) 2 1 2 (cid:21)
For the same tunneling events one finds gη1η2(t)= π.
ll 2 The Klein factor contribution with ǫ = − is simply the
The application of these Klein factor Green’s function
complexconjugateofEq. (15). Thistermis crucialfora
is mostrelevantwhen consideringtime orderedproducts
correct description of the noise cross correlations in the
hTKΠl=j,k eiǫl1√νΘl(tηl1l1).....eiǫlml√νΘl(tηlmlmll)i FQHE. For ν = 1, it allows to recover the negative cor-
relation result obtained with scattering theory [14]. For
=exp −ν ml ml′ ǫlkǫl′k′glηll′kηl′k′(tlk−tl′k′) (12) bνe=ha1v/io3r,,twhehinchoisisewcoerarkeelrattihoanns etxhheiibnitteagneracnatsie-b.unching
(cid:0) Xl l′Xk=1kX′=1 (cid:1)
≤ Finally,weturntothecomputationofnon-equilibrium
(l,l =j,k) which vanishes unless ml ǫ =0. averagesattachedtoagivenlead: thezerofrequencycur-
′ k=1 lk
ConsiderfirstthecaseofatwoedPgesystem,whereonly rent fluctuations in lead 1, denoted S˜j(ω = 0). Pertur-
one Klein factor is necessary. In Eq. (12), the tunnel lo- bationtheorytells usthatthere aretwocontributionsto
cations are l = l = 1, so that the only Green’s function the current and noise:
′
3
hI i=hI i +hI i (16) gate, and the whole system requires once again three
j j 0 j K
S˜ =S˜0+S˜K (17) edges.
j j j
The current and the noise contain an “exact” contri-
hIji0 and S˜j0 invoke only the tunneling operatorsTj and butionwhereallothertunnelingprocessesarediscarded,
Tj† between edges 0 and j. As discussed above, these togetherwithKleinfactorcorrections. Thislattercontri-
terms do not involve Klein factors as they couple only bution illustrates how current and noise are affected by
two edges, and they thus correspond exactly to the cur- the presence of additional stochastic contributions. It is
rent and noise in a two terminal system derived in [15]: symptomatic of a tunneling process (T2) which is influ-
enced by another via a non linear coupling (T ): in Fig.
1
+
hI i ≡−ev ∞dθρ(θ)|S (θ−θ )|2 1, the fluctuations at 1 propagate to the right along the
j 0 F + B
Z − bottom edge, affecting all transport properties at 2. In
νe −∞ dhI i the language of quantum dissipative systems, it can be
S˜j0 ≡ 2(1−ν)(cid:20)V dVj 0 −hIji0(cid:21) (18) mapped to the problem of a brownian motion [17] in a
periodic potential with two coupled particles.
with ρ(θ) the kink density, S (θ−θ ) the kink trans-
+ B T.M. acknowledges support from the LSF program at
−
mission probability and the barrier height appears via
the Brown Submicron Center of Weizmann Institute.
the kink parameter θ . Its leading order is proportional
B
to |Γ |2, the Poisson limit. On the other hand, the sec-
j
ondcontributionofEq.(17)involvesadditionaltunneling
processes between the injecting edge 0 and other edges
k, for instance k > j. In perturbation theory, these ad-
ditional terms are proportional to |ΓpΓp′|2 (p,p ≥ 1)
j k ′ [1] R. B. Laughlin, Rev. Mod. Phys. 71, 863 (1999); H.L.
and they signal the possibility that quasiparticles from
Stormer, ibid,875 (1999).
an external “reservoir” (edge k) are exchanged with the
[2] C.L.KaneandM.P.A.Fisher,Phys.Rev.Lett.72,724
injecting edge 0. The leading order |Γ Γ |2 correctionto
j k (1994).
the Klein factor contribution reads:
[3] C. de C. Chamon, D. E. Freed, and X. G. Wen, Phys.
2|Γ Γ |2 Rev. B 51, 2363 (1995-II).
S˜jK(ω =0)= h4jα4k ηXη1η2Xǫǫ′ η1η2Z dtZ dt1Z dt2 [4] LP.hySsa.mRineva.daLyeattr., 7D9.,C2.52G6la(t1t9l9i,7)Y;.RJ.ind,e-aPnidccBio.ttEotieetnanle.,,
eiω0[ǫt+ǫ′(t1−t2)]e2ν[Gη−η(0,t)+Gη1η2(0,t1−t2)] [5] VN.atJu.reG3o8ld9m, 1a6n2a(n1d997B).. Su, Science 267, 1010 (1995);
× eνǫǫ′[Gηη2(−a,t−t2)+G−ηη1(−a,−t1)]eiξ′ −1 (19) V. J. Goldman, I. Karakurt, Jun Liu and A. Zaslavsky,
(cid:16)eνǫǫ′[Gηη1(−a,t−t1)+G−ηη2(−a,−t2)] (cid:17) [6] Ic.onSdafi-m, Pat./D00e1v0il0la9r3d. and T. Martin, Phys. Rev.Lett. 86,
Where the Klein factor time ordered product reads: 4628 (2001).
[7] Hanbury-BrownandQ.R.Twiss,Nature177,27(1956).
eiξ′ =hTKe−iǫ√νθj(tη)eiǫ√νθj(0−η)e−iǫ′√νθk(tη11)eiǫ′√νθk(tη22)i [8] M. Fabrizio and A. Gogolin, Phys. Rev. B 51, 17827
=eνπeiνπǫǫ′((η+η2)sgn(t t2)+( η+η1)sgn( t1))/4 (1995);J.vonDelftandH.Schoeller,AnnalenderPhysik
− − − 4, 225 (1998); C. Nayak et al., Phys. Rev. B 59, 15694
eiνπǫǫ′(−(η+η1)sgn(t−t1)+(η−η2)sgn(−t2))/4 (20) (1999).
[9] X.G. Wen, Int. J. Mod. Phys. B 6, 1711 (1992); Adv.
Thecalculationofthe currentanditscontributionhI1iK Phys. 44, 405 (1995).
linked to the Klein factors follows the same guidelines,
[10] M.P.A.FisherandL.I.GlazmaninMesoscopic Electron
except that the Keldysh evolution operator in Eq. (5) is Transport, L.Kouwenhoven,G. Sch¨onand L. Sohneds.,
expanded to third order. NATO ASI Series E345, Kluwer Ac. Publ., Dordrecht
In conclusion, a systematic procedure is provided for (1997).
computing transport quantities such as current, noise, [11] See,forinstance,H.Kleinert,inPath Integrals inQuan-
noise correlations in multiple FQHE edges. Generaliza- tum Mechanics, Statistics and Polymer Physics, pp. 139
tions to hierarchical fractional Hall states and to multi- (World Scientific1995).
[12] S. Kettemann Phys. Rev.B. 55, 2512 (1997).
quasiparticle tunneling are envisioned. Multiple quan-
[13] M. Hennyet al.,Science 296, 284 (1999).
tum Hall edge transport problems are necessary to ad-
[14] T. Martin and R. Landauer, Phys. Rev. B 45, 1742
dress the nature of the statistics of the underlying exci-
(1992); M. Bu¨ttiker, ibid.46, 12485 (1992).
tations: the computation of the noise cross correlations
[15] P. Fendley, A. Ludwig, and H. Saleur, Phys. Rev. Lett.
requires Klein factors. Multiple edges allow to study 74, 3005 (1995); ibid. 75, 2196 (1995).
fractional quantum transport through a constriction of [16] M. Heiblum,private communication.
apreparedfluctuatinginjectingcurrent[16]. Thisinject- [17] A.Schmid,Phys.Rev.Lett.51,1506(1983); F.Guinea,
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4
