Colored noise in the fractional Hall effect: duality relations and exact results

14 Pages·0.23 MB·English

by  Claudio Chamon

Colored noise in the fractional Hall effect: duality relations and exact results
Claudio Chamon1 and Denise E. Freed2
1 Department of Physics, Boston University, Boston, MA 02215
2 Schlumberger - Doll Research, Old Quarry Road, Ridgefield, CT 06877
9 Westudy noise in theproblem of tunnelingbetween fractional quantumHall edgestates within
9 a four probe geometry. We explore the implications of the strong-weak coupling duality symme-
9 try existent in this problem for relating the various density-density auto-correlations and cross-
1
correlations between the four terminals. We identify correlations that transform as either “odd”

n or “anti-symmetric”, or “even” or “symmetric” quantities under duality. We show that the low
a frequencynoiseiscolored,andthatthedeviationsfromwhitenoiseareexactlyrelatedtothediffer-
J entialconductance. Weshowexplicitlythattherelationshipbetweentheslopeofthelowfrequency

4 noise spectrum and the differential conductance follows from an identity that holds to all orders
1 in perturbation theory, supporting the results implied by the duality symmetry. This generalizes
the results of quantum supression of the finite frequency noise spectrum to Luttinger liquids and
]
fractional statistics quasiparticles.
l
l
a
PACS: 73.40.Hm, 71.10.Pm, 73.40.Gk
h
-
s
e I. INTRODUCTION ρRI ρRO
m
VG
.
t
a
m R
- It x
d
n L
o Measurements of current fluctuations in a system can
c yield much information about its excitation spectrum.
[ Thishasbeenshowntobethecasefortunnelingbetween VG
ρ ρ
1 edge states of fractional quantum Hall (FQH) liquids. LO LI
(a)
v Two experimental groups, one in Saclay [1] and another
8 at the Weizmann Institute [2], have recently been able
2 to measure the shot noise level in the tunneling current ρRI ρLO
1
1 between FQH edges. The results of the experiments are VG
consistent with the interpretation of tunneling of frac-
0
9 tionally charged quasiparticles. The geometries for such
R
9 measurements are shown in Fig. 1, and studies of var- It
/ ious properties of the noise spectrum have been carried x
t
a out recently [3–9].
m L
- For a small tunneling currentIt between the FQH liq-
d uid edges, the shot noise level should approach the clas- VG
n
sical limit 2e I , where e is either the Laughlin quasi- ρ ρ
o ∗ t ∗ RO LI
particle charge (e = νe) for the geometry in Fig. 1a, (b)
c ∗
: or the electron charge (e∗ = e) for the geometry in Fig.
v
1b[3,4,7]. Inthisclassicallimit,thetunnelingeventsare FIG.1. Twogeometriesfortunnelingbetweenedgestates.
i
X uncorrelated and the noise spectrum at low frequencies In (a) quasiparticles can tunnel from one edge to the other.
r appear to be white or frequency independent. As the In (b) only electrons can tunnel across. The subscripts for
a tunneling current increases, two different effects become the densities ρ in the four terminals determine the chirality
manifest. First, the zero-frequency level deviates from (R,L) of the branch and whether the branch is incoming to
the classical level [4,5,9]. Secondly, the low frequency (I) or outgoing from (O) thetunnelingpoint.
spectrum is no longerwhite, and developsa cusp at zero
frequency [6,7]. This colored noise structure offers the Ithasbeenknownforawhilethatstatisticsplayarole
possibility to investigate how the results known for non- in the suppression of shot noise. In the case of trans-
interacting particles with Fermi statistics are modified mission of electrons through a quantum point-contact
by the correlation effects in FQH liquids, which contain (QPC), with transmission coefficient T, the zero fre-
excitations with fractional charge and statistics. quency shot noise level is given by 2eI(1 T), where
−
1
I = e2T V is the current transmitted across the point Chiral Luttinger liquids are realized on the edges of
h
contact. Theeffectsofthefermionicstatisticsistoreduce fractional quantum Hall liquids. We will study the low-
by a factor 1 T the classical shot noise level 2eI [10]. frequency slope of the noise spectrum for a four probe
−
The implications of quantum statistics, however,are not geometry. This consists of looking at density fluctua-
limitedsolelytothezerofrequencynoiselevel. Thenoise tions in the left (L) and right (R) edges, both incoming
spectrum is not white, so that the zero-frequency noise (I) to and outgoing (O) from the tunneling point. We
level alone cannot describe the full frequency dependent will show that correlations between densities in pairs of
noise spectrum. terminals are related to the transmission and backscat-
For non-interacting electrons, the excess noise spec- tering differential conductances in ways dictated by the
trum S (ω), defined as the difference between the non- strong-weak duality symmetry present in the problem.
ex
equilibrium (V = 0) and equilibrium (V = 0) noise, is
6
given by The paper is organized as follows. In Section II we
summarize our results for the relationships between the
e2 slope of the noise spectra and the differential conduc-
S (ω)= T(1 T) (ω ω ) θ(ω ω ) , (1)
ex π − J −| | J −| | tances. In Section III we discuss the four terminal ge-
ometry for measurement of auto and cross-correlations
where ωJ = eV/¯h is the “Josephson” frequency set by between pairs of terminals, and we derive in detail the
the applied voltage. Notice that the excess noise de- relationships for the correlations of voltage/currentfluc-
creases linearly with frequency from the zero-frequency tuations between pairs of terminals, which follow from
shotnoiselevelSex(0)=2eI(1 T)tozeroattheJoseph- the dual descriptions of the problemin terms of electron
−
son frequency ωJ, remaining zero beyond this frequency or quasiparticle tunneling (Fig. 1). By using current
scale. Although it is hard to probe experimentally the conservationandanothersymmetryoperation,whichex-
noise spectrum both at high-frequencies near ωJ and changesrightRandleftLedgesandreversesthevoltage,
at low-frequencies (due to 1/f noise), it is possible to weareabletorelatecorrelationsinonepicturetothosein
observe the spectrum at intermediate frequencies. In- thedual. Wethenshowhowtheserelationstie,inpartic-
deed,measurementsofthespectruminthisintermediate ular,theslopeofthespectratodifferentialconductances
range have been recently obtained by Reznikov et. al. in the problem. We derive these relationships between
[11]. These measurements can be extrapolated to zero noisespectraslopeanddifferentialconductancesdirectly
frequency, yielding the experimental observation of the fromtheboundarysine-Gordonmodelthatdescribesthe
quantum suppression of shot noise in a QPC by a factor tunneling problem in Sections IV (for auto-correlations)
1 T. Hopefully, further refinements of the experimen- and V (for the tunneling current). We show that the re-
−
tal technique would allow the measurement of the slope lationsholdtoallordersinpertubationtheory,signaling
∆Sex = e2 T(1 T) of the excess noise spectrum, thus the existence of an exact identity. We conclude the pa-
∆ω −π −
probing quantum effects for finite frequencies and show- per in Section VI with a discussion of our results, and a
ing that the excess noise spectrum is not white. comparisontoresultsonthe lowfrequencyspectrumob-
It is noteworthy that the slope of the noise spec- tained from the thermal Bethe ansatz and form factors.
trum near zero frequency keeps a close relationship to In the firstreading of the paper, we suggestthat readers
the transport coefficient T. This relationship is not peruse Section VI prior to Sections IV and V.
the one supplied by the fluctuation dissipation theorem
(notice that the slope dependence on T is quadratic).
One can also find the relationship between T and the
slope of the full noise spectrum S(ω)= S (ω)+S (ω)
ex eq
(excess plus equilibrium contribution to noise); using
S (ω)= e2T ω , one finds that
eq π | |
∆S e2 II. SUMMARY OF RESULTS
= T2 . (2)
∆ω π
Therelationconnectingtheslopeofthenoisespectrum
at low-frequencies to the transportcoefficient T through
the point contact, shown above for non-interacting elec-
trons, can be generalized. In this paper we will general-
izesucharelationtothecaseoftunnelingbetweenchiral
Luttinger liquids. These strongly correlated states have In this section we will give a brief summary of some
excitations that carryfractionalquantum numbers, such of the main results and ideas in this paper. We begin by
as charge and statistics, which make them ideal candi- describing the low-frequency structure of the noise spec-
dates for the study of the effects on quantum noise due trum for the tunneling or backscattering current It be-
to generalized charge and statistics. tweenthe edges ofthe FQHliquid, asdepicted in Fig. 2.
2
S (ω) A =2g2 . (5)
t T T
These results are valid in both the picture in Fig. 1a,
colored noise
where quasiparticles tunnel, or in the picture in Fig. 1b,
2
slope g where electrons tunnel. We indicate which picture we
t
usebyaq oresuperscriptinthequantityofinterest,for
shot noise
level example, Aq,Ae and gq,ge.
t t t t
This brings into focus another point we discuss in this
low freq. high freq.
paper: how different quantities transform between the
ω quasiparticle and electron tunneling pictures. For exam-
ple, it is widely known that
FIG. 2. Low frequency noise spectrum for the tunneling ge =1 gq , (6)
current It between edges. t − t
which can be interpreted as a duality relation. This
comes about because the two pictures describe the same
The noise in the backscattering current is defined as
physical system, but one is the strong tunneling limit of
the other and vice-versa. Eq. (6) then simply follows
S (ω)= dt cos(ωt) I (t),I (0) .
t t t from the definitions of I and I in the two pictures of
h{ }i t T
Z Fig. 1. These two pictures are actually dual in another,
So far, much of the theoretical study of noise in the stronger, sense: the model for the two pictures is self-
fractional quantum Hall effect has focused on the zero- dual, i.e., the tunneling mechanism for the two pictures
frequencyshotnoiselevelorS (ω =0). Forsmalltunnel- is described by the same Lagrangian, but with a differ-
t
ing currents the shot noise approaches the classical level ent sets of parameters (up to counterterms required for
St(ω = 0) = 2e∗It. The solution for the zero-frequency renormalization). The parametersare the tunneling am-
shotnoiseisknownexactly,viatheBetheansatz,forany plitude Γ and the Luttinger parameter g. In the quasi-
value of the applied voltage V and tunneling amplitude particle tunneling picture they take the values Γ and g,
q
Γ. However, the finite ω results are rather more com- and in the electron tunneling picture they take the val-
plicated, and require knowledge of form factors in order ues Γ and g 1, respectively. This allows us to write the
q −
to calculate the current-currentcorrelations defining the conductances in the two pictures using a single function
noise spectrum [6]. In this paper we introduce an alter- g :
t
native approach.
The spectrumatlow frequencies is not white, i.e., fre- gq =g (g,Γ ) (7)
t t q
quency independent or flat. The frequency dependence ge =g (g 1,Γ ) . (8)
or color of the noise can be described in terms of the t t − e
slope or derivative of the noise spectrum with respect to
Because the backscattering conductances gq and ge in
the frequency: t t
the two pictures are related to each other by Eq. (6) in
1 S (ω) S (0) such a simple way, we can ask whether other physical
t t
At = lim − , (3) quantities in the two pictures have similar relationships.
GH ω→0 ¯h|ω| The coefficients Aq and Ae do not transform in a such a
t t
wherewehavedividedtheslopebytheHallconductance simple way. Instead, we will show that other quantities,
G = νe2/h in order to make A a non-dimensional suchasthefourterminalnoisecorrelations,dohavesim-
H t
quantity. pletransformationlaws,althoughtheydonotnecessarily
In this paper we will show that the slope A is di- satisfythesametransformationrulesastheconductance.
t
rectly related to the differential backscattering conduc- Under the duality transformation, cross-correlations be-
tance G =dI /dV by tweenanincomingandout-goingbranchtransforminan
t t
“odd” or “antisymmetric” way like the backscattering
G 2 conductance. In contrast, the auto-correlations trans-
A =2 t =2g2 , (4)
t G t form in an “even” or “symmetric” way. Specifically, we
(cid:18) H(cid:19) show that the derivative with respect to frequency of
where we also define a dimensionless differential conduc- these noise correlations (as defined below) satisfy
tance g . The relation in Eq. (4) is a generalization of
t
theresultfornon-interactingelectrons(Luttingerparam- Aq =1 Ae Aq =Ae . (9)
cross − cross auto auto
eter g = 1) to correlated chiral Luttinger liquids. Sim-
ilarly, we can relate the noise in the transmission cur- We point out that A and A have both even and odd
t T
rent,S (ω), to the differentialtransmissionconductance components, which is why they do not transform in a
T
G =dI /dV (with g =G /G ): simple way.
T T T T H
3
ItistemptingtoinferfromEqs.(6)and(9)theansatz completely,andweobtainthe secondpicture,Figure1b,
A = g or A = g . (There are two combina- which is described by electron tunneling with small Γ .
cross t cross T e
tions of cross correlations between incoming and outgo- ThusthelargeΓ limitofthequasiparticlepictureshould
q
ing branches,depending onwhether their chiralityis the be the same asthe smallΓ limit ofthe electronpicture,
e
sameoropposite). Theseansatzsatisfythecorrecttrans- and vice versa. In other words, the two pictures should
formation under duality. Indeed, we show in this paper both describe the same physical system.
thatthisansatziscorrect. Thisissupportedbyapertur- Thismeans thatthe incomingandoutgoing,rightand
bative calculation to all orders. The calculation is done left moving densities in the two pictures of Fig. 1 are
explicitly using the self-dual boundary sine-Gordon the- related by
ory that describes the tunneling mechanism in both the
ρq =ρe (10)
quasiparticle and electron tunneling pictures. RI RI
The results for the slopes A ,A and A can all ρq =ρe (11)
cross t T LI LI
be expressed directly in terms of the differential con- ρq =ρe (12)
RO LO
ductance, bypassing the non-linear series expansion in ρq =ρe , (13)
terms of the voltage V and the tunneling amplitude Γ. LO RO
This is also the case for Aauto, which we find to be where the subscripts I, O denote incoming andoutgoing
Aauto = 1 2gtgT = 1 2gt(1 gt). In general, we can branches,andR,Ldenoteright-movingandleft-moving.
− − −
always write a quantity like Across as a function f(g,gt) Thesuperscriptq oremeansthefunctionforthedensity
of both the Luttinger parameter g and the differential is given in the quasiparticle or electron picture, respec-
conductance gt. However, our results indicate that the tively (see Figure 1). As described above, the densities
differential conductance gt is the single parameter con- inthequasiparticlepicturearefunctionsofg andΓq and
trolling the behavior of the slope of the noise. the densities in the electron picture are functions of g 1
−
and Γ .
e
Theseequationscanbeusedtorelatethecurrentsand
III. DUALITY RELATIONS correlations in the quasiparticle picture to those in the
electron picture. In order to solve for these correlations,
we need further constraints on them. These constraints
Inthissection,wewillshowhowthesymmetriesofthe
will take the form of a relation between the correlatorin
system,whichincludeavoltagereversalsymmetryanda
onepictureasafunctionofonesetofparametersandthe
duality symmetry, combined with current conservation,
same correlator in that picture as a function of another
lead to identities among the different correlators. We
set of parameters. We will find that these relations will
then make use of these identities and one further ansatz
greatly restrict the form of the correlators, but will not
aboutthedependenceofthenoiseonthedifferentialcon-
determine them uniquely.
ductance to solve for the noise in the Hall and tunneling
To derive these relations, we must make use of addi-
currents in terms of differential conductances.
tional properties of the system. The first one is current
We begin by describing two dual pictures of the sys- conservation,whichsimplystatesthat,inagivenpicture,
tem. In the first picture, the constriction is not pinched
off,asinFigure1a,sothequasiparticlescantunnelfrom ρ +ρ =ρ +ρ . (14)
RO LO RI LI
one edge to the other. The charge of the particle that
(Strictly speaking, except right at the impurity, this
tunnelsise . ItstunnelingamplitudeisΓ ,andtheLut-
∗ q
equation is purely classical. Once these densities appear
tinger parameter is g. One may view g as a parameter
in expectation values for the noise, the correlators are
controlling the influence of a tunneling event on subse-
modified by additional phases eiωx, where x is the dis-
quent ones. In the second picture, the constriction is
tance along the edge between the terminal in question
completely pinched off, so there is no longer any quan-
and the impurity). The second property is that quanti-
tumHallliquidinthe centralregion. Nowonlyelectrons
tiesthatarequadraticinthedensities,suchasthe noise,
can tunnel from one edge to the other, so in this picture
are symmetric under inversion of applied voltage
the chargeof the particle that tunnels is e. Its tunneling
amplitude is Γ , and the Luttinger parameter is g 1.
e − V V .
There are two ways in which these pictures are dual →−
to one another. We will beginby describingthe first one This implies that we can exchange the labels R and L
here, and save the second until later in this section. The without changing the value of the density-density corre-
two pictures are dual to one another in the sense that lations. To see this, just rotate the sample by 180 de-
thestrongtunneling limitofoneshouldalsodescribethe grees,andinvertthevoltage: thesesymmetryoperations
weak tunneling limit of the other. For example, as the exchange the labels R and L.
constriction in Figure 1a is narrowed, it becomes easier Next, it is useful to define some of the currents and
for quasiparticles to tunnel from one edge to another, so noisecorrelatorsintermsofthe densities. Thetransmis-
Γ increases. As the constriction is narrowedfurther (so sioncurrentisgivenbyI = ρ ρ andthetunnel-
q T RO LI
h − i
Γ is increased some more) at some point it pinches off ingorbackscatteringcurrentisgivenbyI = ρ ρ .
q t RI RO
h − i
4
It is also useful to define I = ρ ρ , which is the Inparticular,thereisasecondsenseinwhichwemean
H RI LI
h − i
Hall current in the absence of tunneling or backscatter- the two pictures are dual: we assume that the tunneling
ing. isdescribedbyaself-dualtheory,bywhichwemeanthat
For the noise, we will define, for example, the description of the tunneling mechanism is the same
in both pictures. Pictorially, what this means is that
S (ω)= dtcos(ωt) ρ (t),ρ (0) (15) we can reverse the shaded and unshaded regions in Fig-
RI,LO RI LO
h{ }i ure 1b so that it looks exactly like Figure 1a, rotated
Z
by 90 degrees. This signifies that now in both pictures
The other noise correlators, for instance S and
RO,LO
the tunneling should be described in the same way, just
S , are defined likewise. (In general, we will de-
RO,RO with differing parameters – g 1, e, and Γ , or g, e and
note by S the correlator between the terminals α,β, − e ∗
α,β
Γ – depending on whether quasiparticles or electrons
where α and β can take on the values RO,LO,RI,LI). q
are tunneling. (However, the filling fraction, ν, of the
Notice that we dropped thex dependence of the correla-
shaded region remains the same in both cases.) By us-
tors. Wedosobecauseweareprimarilyinterestedinthe
ing the Luttinger liquid framework for both the electron
spectrum at low frequencies, in which case the x depen-
dence canbe neglectedaslong asω x 1. Ifnecessary, tunneling of Figure 1b and the quasiparticle tunneling
−
≪ of Figure 1a, we are implicitly making this assumption.
to distinguish which picture we are considering we will
(However, once questions of renormalization arise and
use the superscripts e and q.
We define the noise S(0) as the correlatorbetweentwo counter-terms must be added to one picture and not the
other,itisnolongerguaranteedthatthesystemreallyis
right or two left moving densities in the absence of the
described by a self-dual theory.) Mathematically, in this
couplingΓbetweenthe RandLbranchesinaparticular
frameworkwe use the same Lagrangian,just with differ-
picture. It is given by
entvaluesofchargeandtunnelingamplitude. Giventhis
S(0)(ω) = ν ω . secondtypeofduality,wecanreplaceSαe,β andSαq,β bya
2π| | single function, S (g,Γ), so that Se = S (g 1,Γ )
α,β α,β α,β − e
We note that SLI,LI = SRI,RI = S(0) in the pres- and Sαq,β = Sα,β(g,Γq), (and similarly for the currents).
ence of any tunneling because the incoming channels The identities between the noise and currents then be-
have yet to be affected by the tunneling. Similarly, come
S = S = 0 since the two incoming channels
LI,RI RI,LI I (g,Γ )=I I (g 1,Γ ) (23)
are completely uncorrelated. t q H − t − e
Itfollowsfromcurrentconservationandthesymmetry S(0) SRI,LO(g,Γq)=SRI,LO(g−1,Γe) (24)
−
undervoltageinversion(R L)that,inagivenpicture, S (g,Γ )=S (g 1,Γ ). (25)
↔ RO,LO q RO,LO − e
I =I I (16)
T H − t Noweachoftheseequationsisaduality relationrelating
S =S =S(0) S =S(0) S (17) a function at one set of parameters to the same func-
RI,RO LI,LO RI,LO LI,RO
− −
S =S =S(0) S . (18) tion at another set of parameters. This kind of rela-
RO,RO LO,LO − RO,LO tiongreatlyrestrictsthepossibleformofthecurrentand
Theseequationsrelateonesetofcurrentsornoiseinone noise.
picture to another set of currents or noise in the same Tofindtheformofthenoisethatissuggestedbythese
picture. Next, we can use the relations between the two duality relations,we begin by noting that Eq.(23)forIt
pictures to write expressions for the current or noise in and Eq. (25) for SRI,LO have a very similar form, which
onepictureintermsofthesamecurrentornoisefoundin indicates there may be a simple relation between the
the other picture. Combining Eqs. (16-18) and (10-13), function that satisfies the duality relation for the cur-
we find rentandtheonethatsatisfiesthedualityrelationforthe
cross-correlationS . Onemusttakecare,though,in
RI,LO
Itq =IH −Ite (19) trying to equate It and SRI,LO because they have differ-
Se =Sq =S0 = ν ω (20) entdimensionsandoneisafunctionofωandtheotheris
RI,RI RI,RI 2π| | not. Instead, we will look at the slope ARI,LO of SRI,LO
S(0) Sq =Se (21) near ω =0, which we will normalize as follows:
− RI,LO RI,LO
SRqO,LO =SReO,LO. (22) A = lim SRsinI,gLO. (26)
Because these equations relate the noise in one picture RI,LO ω 0 S(0)
→
tothenoiseinthe otherpicture,wecannotmakefurther
In this equation, Ssing is the part of S that is
use of these equations unless we either know what the RI,LO RI,LO
singular as ω 0. We will also define the dimensionless
noise is in one of the two pictures, or we know another →
differential conductance as
relation between the noise in the two different pictures.
If we make anadditionalassumption, we can obtainthis 1 dI
t
g = , (27)
second set of relations. t G dV
H
5
where G = νe2/h. Then the duality relations for For g =1/2,this can be written in terms of the differen-
H
A and g become tial conductance as follows:
RI,LO t
g (g,Γ )=1 g (g 1,Γ ) (28) A =2(g g2). (33)
t q − t − e RO,LO t− t
A (g,Γ )=1 A (g 1,Γ ). (29)
RI,LO q RI,LO − e
− Giventhe duality relationfor g , this expressionsatisfies
t
Thus the dimensionless conductance and the normalized the duality relation for ARO,LO. As before, ARO,LO is a
slope of the noise satisfy exactly the same duality rela- powerseriesin(V/TB)(2g−2) startingwiththe firstorder
tion. term, so it is given by
To look for a simple relationbetween the conductance
andA ,weconsiderthecasewhentheLuttingerpa- ∞
RI,LO A = b (g)gn. (34)
rameter g =1/2. In this case, there is an exact solution RO,LO n t
n=1
for the noise. Guided by the fact that the conductance X
and the slope of the noise satisfy the same duality rela- If againwe assumethe b are independent ofg, then the
n
tion, we find we can rewrite the expression for ARI,LO, value of ARO,LO at g =1/2 determines bn for all g, and
calculated in reference [8], in terms of the conductance. the slope of the noise has the form given in Eq. (33).
It is given by Combiningthis resultwith Eq.(18),whichrelatescorre-
lations between pairs of outgoing branches,we canwrite
A =g . (30)
RI,LO t A =1 A , or
RO,RO RO,LO
−
Thusweseethatinthiscasethereis,indeed,asimplere-
A =A =1 2g g . (35)
lation between ARI,LO and the differential conductance, auto RO,RO − t T
which begs the question of whether this is true for all g.
This is our conjecture for the auto correlations in the
InSectionIV,wewillexplicitlycalculatetheslopeofthe
outgoing branches.
noisetoallordersinperturbationtheory. Thisperturba-
Finally,weconcludethissectionbystatingthe conjec-
tiveexpansionisvalidwhen(V/T )(2g 2) issmall,where
B − ture for the slope of the noise in the tunneling current,
T Γ 1/(g 1). However, if we want to know the value
foBr ∝(V/T− )(2−g 2) large, we can use the duality relation At, and in the transmission current, AT. The slope of
B − the tunneling noise is defined as
and calculate in the dual picture. In this way, we obtain
the value of ARI,LO over the whole region of parameter St
space. We find that Equation (30) does hold for all g. At =ωlim0S(0), (36)
Here, instead, we will show how relation (30) follows →
from one additional ansatz. Coloumb gas expansions where Ssing is the singular part of the tunneling noise,
t
for the conductance and noise are both power series in and A is defined similarly. Using the definition of the
T
(V/TB)(2g−2) beginning with the term (V/TB)(2g−2). As transmission and tunneling currents, we find
aconsequence,wecanalwaysexpressA asapower
RI,LO
series in gt, given by At = ARO,LO+2ARI,LO (37)
−
A = ∞ a (g)gn, (31) and
RI,LO n t
n=1
X A =2 A 2A (38)
T RO,LO RI,LO
− −
where a (g) depends on the Luttinger parameter g. Be-
n
cause both A and g satisfy the duality relation, The noise in the transmission current in one picture
RI,LO t
thisputssomerestrictionsonthea ,butdoesnotdeter- equals the noise in the tunneling current in the other
n
mine them uniquely. However, if we make the assump- picture. However, in a given picture, there is no simple
tion that an does not depend on g, then we can use the duality relation for At or AT. Instead, our solutions for
solution at g = 1/2 to fix the an, with the result that ARO,LO and ARI,LO imply that
A =g for all g. (This assumption is equivalent to
RI,LO t
theonemadein[9]thatenabledWeisstousetheduality At =2gt2, (39)
relation to solve for the conductance. Since A sat- A =2g2. (40)
RI,LO T T
isfies the same duality relation as the conductance and
equalsthe conductanceforg =1/2,hisansatzandensu- In Section V, we use a multipole expansion to calculate
ing calculation also uniquely determine A ). the tunneling noise to all orders in perturbation theory,
RI,LO
We can use the same line of reasoning for the slope of andfindthatEquation(40)does,indeed,hold. Thus,we
S , which is defined as expect to find a simple relation between the slope of the
RO,LO
noise at low frequency and the square of the differential
Ssing conductance. This has the same formas Shiba’s relation
RO,LO
A = lim . (32)
RO,LO ω 0 S(0) for the dissipative two-level system [12,13].
→
6
IV. CROSS-CORRELATIONS
In the previous section we related the slopes of different noise correlations to the differential conductances by
exploring the duality symmetry in the problem. We used an ansatz that the coefficients for the series expansions
of these slopes (as a power series in the conductances) did not depend on the Luttinger parameter. In this section
we formally justify this ansatz by explicitly calculating, starting from the boundary sine-Gordon Lagrangian, the
noise cross-correlations between an incoming branch and an outgoing branch. In other words, here we show that
the independence of the expansion coefficients on the Luttinger parameter g is a specific property of the boundary
sine-Gordon Lagrangianthat describes the tunneling problem.
The Lagrangianthat describes the tunneling between chiral Luttinger liquids through a QPC is
= + +Γ δ(x) eiωJt ei√g(φR(t,0)+φL(t,0)) , (41)
R L
L L L
where = 1 ∂ φ ( ∂ ∂ )φ is the Lagrangian for the free chiral bosons. We will calculate the zero-
LR,L 4π x R,L ∓ t − x R,L
frequency singularity in the noise spectrum to all orders in a perturbative expansion in Γ.
We will show that the slope of the spectrum for the cross-correlations is linearly related to the differential trans-
mission and backscattering conductances. We proceed in the following way: we will expand the density-density
correlations for the cross noise to all orders in the tunneling amplitude Γ, and compare them to the expansion, also
to all orders, of the differential conductances.
We write the correlations between density operators as follows:
ρ (t,x )ρ (0,x ) , (42)
a 1 b 2
h i
where a,b take the values +1 for R moving branches and 1 for L moving ones. Such compressed notation makes
−
it simpler to identify incoming and outgoing branches in a unified way for both left and right movers: ρ (t,x ), for
a 1
example, is the density in an incoming or outgoing branch if ax <0 or ax >0, respectively.
1 1
The densities are related to the fields φ through ρ = √ν∂ φ , so that we can write
R,L R,L 2π x R,L
ν
hρa(t,x1)ρb(t′,x2)i= (2π)2 ∂x1∂x2hφa(t,x1)φb(t′,x2)i , (43)
where it is convenient to use
d d
hφa(t,x1)φb(t′,x2)i= dλ1dλ2heiλ1φa(t,x1) e−iλ2φb(t′,x2)i λ1,λ2=0 . (44)
(cid:12)
The last correlation function is easy to calculate perturbatively using (cid:12)
Tc(eiλ1φa(t,x1) e−iλ2φb(t′,x2)) = 0 Tc(S( , ) eiλ1φa(t,x1) e−iλ2φb(t′,x2)) 0 , (45)
h i h | −∞ −∞ | i
where 0 is the unperturbed ground state, and T is the ordering along the Keldysh contour (see Fig. 3 and Refs.
c
| i
[7,8]). The scattering operator S( , ) takes the initial state, evolves it from t = to t = and back to
−∞ −∞ −∞ ∞
t= . The use of the Keldyshcontour is necessary in the treatment of non- equilibrium problems, suchas the one
−∞
we have in hand. A more detailed description of the method in the context treated here can be found in Ref. [7].
top
bottom
FIG. 3. Keldysh contour for the non-equilibrium Coulomb gas expansion. Time evolves forward in the top part of the
contour, and backwards in the bottom. Charges from the Coulomb gas expansion are inserted in both the top and bottom
pieces.
7
TheperturbativetreatmentcorrespondstoaCoulombgasexpansion[7]. Thenonzerocontributiontothecorrelation
above comes from the neutral terms in the expansion, thus only even orders in Γ contribute. To (2n)-th order, we
have an insertion of n positive charges and n negative charges. We will label the times at which they are inserted in
the expansion t , i = 1,...,2n, with i = 1,...,n corresponding to the + charges (q = +1), and i = n+1,...,2n for
i i
the charges (q = 1).
i
− −
Tc(eiλ1φa(t,x1) e−iλ2φb(t′,x2)) = (46)
h i
2n 2n
∞ (iΓ)n(iΓ∗)n dti ei ni=1ω0(ti−tn+i) 0 Tc eiλ1φa(t,x1) e−iλ2φb(t′,x2) eiqi√gφ(ti,0) 0 ,
nX=0 IciY=1 P h | iY=1 !| i
where φ without subscript stands for the sum φ +φ . The expression above is simplified using
R L
0Tc( eiqj √gφ(tj,xj))0 =e−g2 i6=jqiqjh0|Tc(φ(ti,xi)φ(tj,xj))|0i . (47)
h | | i
Yj P
Substituting it into Eq. (44) we obtain
2n
Tc(φa(t,x1)φb(t′,x2)) = ∞ ( 1)n Γ2n dti ei ni=1ω0(ti−tn+i)e−g2 i6=jqiqjh0|Tc(φ(ti)φ(tj,xj))|0i
h i − | |
nX=0 IciY=1 P P
2n 2n
qi √g 0Tc(φ(ti,0)φa(t,x1))0 qj √g 0Tc(φ(tj,0)φb(t′,x2))0
×(" h | | i#× h | | i
i=1 j=1
X X
 
+ 0T (φ (t,x )φ (t,x ))0 (48)
c a 1 b ′ 2
h | | i)
The last term in the expression above is simply proportional to 0T (φ (t,x )φ (t,x ))0 . The proportionality
c a 1 b ′ 2
h | | i
constant is equal to Z = 0S( , )0 1. This is the zero-order contribution.
h | −∞ −∞ | i≡
Inorderto carryoutthecalculations,weintroducenotationthatkeepstrackofthe positionofthe insertedcharges
along the contour, i.e., whether they are in the forward (or top) branch, or in the return (or bottom) branch (see
Fig.3andRefs.[7,8]). Thepositionofthechargesisimportantforthecomputationofthecontour-orderedcorrelation
function 0T (φ (t ,x )φ (t ,x ))0
c R,L 1 1 R,L 2 2
h | | i
ln δ+i sign(t t )[(t t ) (x x )] , both t and t in the top branch
1 2 1 2 1 2 1 2
− { − − ∓ − }
ln δ i sign(t t )[(t t ) (x x )] , both t and t in the bottom branch
= −ln{δ−i[(t t1−) 2(x 1−x 2)] ∓, 1− 2 } t in th1e top 2and t in the bottom branch (49)
1 2 1 2 1 2
−ln{δ−+i[(t −t )∓(x −x )]}, t in the bottom and t in the top branch.
1 2 1 2 1 2
− { − ∓ − }
The compact notation consists of giving indices to the times which contain the information about which branch of
the Keldysh contour they are on, so that tµ is on the top branch for µ=+1, and on the bottom for µ= 1. In this
−
way, we can compress the correlations to a compact form:
Gab (t ,x ;t ,x )=Gab (t t ,x x )= 0T (φ (tµ1,x )φ (tµ2,x ))0
µ1µ2 1 1 2 2 µ1µ2 1− 2 1− 2 h | c a 1 b 2 2 | i
= δ ln(δ+i K (t t )[(t t ) a(x x )]) , (50)
− a,b µ1µ2 1− 2 1− 2 − 1− 2
where K (t) = sign(t) and K (t) = 1. Again, we have used a,b = 1 for R and L fields, respectively. The
correlatio±n±in Eq.±(48) can be wri±tt∓en, usin∓g this compressed notation, as ±
T (φ (t,x )φ (t,x )) =Gab (t t,x x ) +
h c a 1 b ′ 2 i ++ − ′ 1− 2
2n
g ∞ (−1)n |Γ|2n ∞ µidti ei ni=1ω0(ti−tn+i)Pµ1,...,µ2n(t1,...,t2n)
nX=1 {Xµi}Z−∞iY=1 P
2n 2n
×" qi Ga+aµi(t−ti,x1)#× qj Gb+bµj(t′−tj,x2), (51)
i=1 i=j
X X
 
8
where Pµ1,...,µ2n(t1,...,t2n)=e−g2 i6=jqiqj[G+µi+µj(ti−tj,0)+G−µi−µj(ti−tj,0)]. The factors µi simply keep track of the sign
coming from the integration of thePtimes ti along the contour. Notice that the times t and t′ are taken to be on the
top branch.
Now, let
F (ω;x ,x )= ∞ dt eiωt T (ρ (t,x )ρ (0,x ))
ab 1 2 c a 1 b 2
h i
Z−∞
= ν ∂ ∂ ∞ dt eiωt T (φ (t,x )φ (0,x )) , (52)
(2π)2 x1 x2 h c a 1 b 2 i
Z−∞
which can be easily shown, using Eq. (51), to yield
ν
F (ω;x ,x )= ∂ g˜ab (ω,x x )
ab 1 2 −(2π)2 x1 ++ 1− 2
2n
+ (2νπg)2 ∞ (−1)n |Γ|2n ∞ µidti ei ni=1ω0(ti−tn+i)Pµ1,...,µ2n(t1,...,t2n)
nX=1 {Xµi}Z−∞Yi=1 P
2n 2n
×" qi g˜+aaµi(ω,x1) eiωti#× qj g+bbµj(t′−tj,x2) . (53)
i=1 j=1
X X
 
In this equation, the function g is given by gab (t,x)=∂ Gab (ω,x) and g˜ is the Fourier transform of g; they can
µ1µ2 x µ1µ2
be obtained from Eq. (50):
πia eiωax (sign(ω)+sign(ax)) , µ =+1,µ =+1
1 2
πia eiωax (sign(ω) sign(ax)) , µ = 1,µ = 1
g˜µab1µ2(ω,x)=δa,b× 2−π2iπaiaeiωeiaωxaxθ(θω()−ω)− ,, µµ11 ==−+11,,µµ22 ==−−+11 (54)
1 2
−
We will now take ax <0 (incoming state) and bx >0 (outgoing state), so as to discuss the cross-correlations. In
1 2
this case, g˜+aaµ(ω,x1)=−2πiaeiωax1θ(−ω), for both µ=±1. For small |ω| we have
2n n
q g˜aa (ω,x ) eiωti = 2πa θ( ω) ω (t t )+ (ω2) . (55)
i +µi 1 − − i− n+i O
i=1 i=1
X X
One can thus write
ν
F (ω;x ,x )= δ ω θ( ω)
ab 1 2 a,b
− (2π) −
2n n
a θ( ω) ω νg ∞ ( 1)n Γ2n ∞ µidti (ti tn+i) ei ni=1ω0(ti−tn+i)
− − (2π) − | | −
nX=1 {Xµi}Z−∞iY=1 Xi=1 P
2n
P (t ,...,t ) q gbb (t t ,x ) + (ω2) . (56)
× µ1,...,µ2n 1 2n  j +µj ′− j 2  O
j=1
X
 
We can similarly (and more easily) expand the tunneling current (i.e., the difference in the densities in a branch
before and after the impurity) to all orders in Γ.
2n
It =b( ρb(t′,x2) ρb(t′, x2) ) = ib√νg ∞ ( 1)n Γ2n ∞ µidti ei ni=1ω0(ti−tn+i)
h i−h − i − 2π − | |
nX=1 {Xµi}Z−∞Yi=1 P
2n
P (t ,...,t ) q gbb (t t ,x ) . (57)
× µ1,...,µ2n 1 2n  j +µj ′− j 2 
j=1
X
 
The current is defined as positive flowing from the right to the left edge, hence the factor b in the expression above.
By direct comparison with Eq. (56), one can then write
9
ν dI
F (ω;x ,x )= δ +√νg ab t ω θ( ω)+ (ω2) (58)
ab 1 2 a,b
− (2π) dω − O
(cid:18) 0(cid:19)
The noise spectrum is obtained from F (ω,x ,x ) as follows:
ab 1 2
S (ω;x ,x )=S ( ω;x ,x )= ∞ dt eiωt ρ (t,x ),ρ (0,x )
ab 1 2 ba 2 1 a 1 b 2
− h{ }i
Z−∞
=F (ω;x ,x )+F ( ω;x ,x ) , (59)
ab 1 2 a∗b − 1 2
so that we can finally write
ν dI
S (ω;x ,x )= δ √νg ab t ω + (ω2)
ab 1 2 a,b
(2π) − dω | | O
(cid:18) 0(cid:19)
ν G
= ω δ ab t + (ω2) , (60)
a,b
(2π)| | − G O
(cid:18) H(cid:19)
where G = dIt, i. e., the differential conductance, and G = νe2/h is the quantized Hall conductance. We used
t dV H
above that ω0 =e∗V/¯h, and that e∗ =√νg (in units of e=1). Reinserting back h¯ and e (which were both set to 1),
we can write the result in a more physical way:
S (ω;x ,x )=h¯ ω (δ G ab G ) , (61)
ab 1 2 a,b H t
| | −
the final result for cross-correlations(valid when (ax ) (bx )<0).
1 2
×
Notice that the result above satisfies the strong-weak coupling duality symmetry for the cross-noise. For example,
Se (ω) = Sq (ω) should be satisfied. Using the result calculated above, Se (ω) = h¯ ω (G Ge), and
RI,RO RI,LO RI,RO | | H − t
Sq (ω)=h¯ ω Gq. Now, G Ge =Ge =Gq, so that, indeed, Se (ω)=Sq (ω).
RI,LO | | t H − t T t RI,RO RI,LO
Finally, we can divide the cross-correlations by the equilibrium noise S(0)(ω) = ν ω = h¯ ω G , and cast the
2π| | | | H
result in terms of normalized conductances g =G /G and g =G /G :
t t H T T H
g , a=b
A =δ ab g = T . (62)
aI,bO a,b− t gt, a=b
(cid:26) 6
V. AUTO-CORRELATIONS
In this section, we will complete our calculation of the slope of the low frequency noise by finding the slope of the
noise in the tunneling current A . From A and the cross-correlation calculated the previous section we can find all
t t
the other correlators. Since A comes from the noise in the tunneling current,we will calculate directly the tunneling
t
current-currentcorrelation, given by
∞
S (ω)= dtcos(ωt) I (t),I (0) , (63)
t t t
h{ }i
Z−∞
where the tunneling operator is given by
It(t)=iΓeiω0tei√gφ(t,0) iΓ∗e−iω0te−i√gφ(t,0), (64)
−
and φ(t,0)=φ (t,0)+φ (t,0). Notice that S (ω)=S ( ω), so that any term in S that is linear in ω and analytic
R L t t t
−
mustvanish. Thus,atfirstorder,the noise goesas ω . Also,becauseofthe translationalinvarianceofthe correlator,
we can replace cos(ωt) by eiωt. | |
For comparison, we will once again need the tunneling current I = I (t) , where I (t) is defined in Eq. (64). We
t t t
h i
can use the expression for the tunneling current to obtain an expansion similar to the one in Section IV. In this case
there is one “physical”charge in the Coulombgas which comes from the operator I (t). It is located at time t on the
t
top branch and its charge is q , which can be 1. The remaining inserted charges occur at times t and can lie on
0 i
±
either the topor bottom branch,labeledbyµ , with i=1,...,2n 1. They havechargesq whichare chosenso that
i i
−
the total charge (including q ) is zero. With these definitions, the perturbation series for I is
0 t
10