Signatures of fractional Hall quasiparticles in moments of current through an antidot

0.36 MB·English

by  A. Braggio

Signatures of fractional Hall quasiparticles in moments of current through an antidot
Alessandro Braggio1, Nicodemo Magnoli2, Matteo Merlo1,2, and Maura Sassetti1
1Dipartimento di Fisica, INFM-LAMIA, Universit`a di Genova,
Via Dodecaneso 33, I-16146 Genova, Italy
2Dipartimento di Fisica, INFN, Universit`a di Genova,
6 Via Dodecaneso 33, I-16146 Genova, Italy
0
0
(Dated: March 22, 2009)
2
ThestatisticsoftunnelingcurrentinafractionalquantumHallsamplewithanantidotisstudied
n
inthechiralLuttingerliquidpictureofedgestates. AcomparisonbetweenFanofactorandskewness
a
J is proposed in order to clearly distinguish the charge of the carriers in both the thermal and the
shotlimit. Inaddition,weaddresseffectsoncurrentmomentsofnon-universalexponentsinsingle-
5
quasiparticle propagators. Positive correlations, result of propagators behaviour, are obtained in
2
theshot noise limit of theFano factor, and possible experimentalconsequences are outlined.

]
l PACSnumbers: 73.23.-b,72.70.+m,73.43.Jn
l
a
h
- Introduction - The properties of quasiparticles in the correlations. Thepeculiarbehaviourdrivenbythequasi-
s
e fractional quantum Hall effect (FQHE) have received particles could allow for a direct estimate of possible
m greatattentionsinceLaughlin’sworkforthestatesatfill- renormalizationeffects inpropagators. Thechoiceofthe
. ing factor ν = 1/p, p odd integer, in which gapped bulk systemhasbeenmotivatedbyrecentexperiments[11]on
t
a excitations were predicted to exist and to possess frac- fractional charge and statistics. These geometries seem
m tionalchargee∗ =νe (e= electroncharge)[1]. Atheory indeed a promising candidate to verify experimentally
- of the FQHE in terms of edge states has been proposed our predictions.
d byWen[2]. Thistheoryrecoveredthefractionalnumbers Model -Edgestatesformattheboundariesofthesam-
n
ofquasiparticlesintheframeworkofchiralLuttingerLiq- ple and around the antidot (Fig. 1(a)); tunneling barri-
o
c uids(χLL),andindicatedtunneling asanaccessibletool erscoupletheantidotwithbothedges. TheHamiltonian
[ to probe them [3]. A charge e/3 of quasiparticles in the readsH =H0+H0 +H0 +HAB+HT+HT,wherethe
L R A R L
ν = 1/3 state was indeed measured in shot noise ex- H0areWen’sHamiltoniansfortheleft,rightandantidot
1 l
v periments with point-contact geometries and edge-edge edge (l = L,R,A), HAB ∝jA·A describes the coupling
0 backscattering [4]. In addition, χLL theory predicts a of the antidot current j with the vector potential A,
A
7 universal interaction parameter equal to ν. The result- and HT is the tunneling between the i = L,R infinite
i
5 ing edge tunneling density of states should reflect in a edges and the antidot. With ~=1, one has [12]
1
power-law behaviour of I −V curves with universal ex-
0 v
6 ponents,e.g. ν−1 inthe caseofmetal-edgetunneling [3]. Hl0 = 4πν Z dx(∂xφl(x))2, (1)
0 Experimentswithedgestatesatfilling1/3indeedproved
t/ a power-law behaviour but with an exponent different where v is the edge magnetoplasmon velocity and φl(x)
a from3[5],anddeviationswereobservedalsointhepoint are scalar fields satisfying the equal-time commutation
m contact geometry [4, 6]. The disagreement of χLL pre- relations [φl(x),φl′(x′)] = ±iπνδll′sgn(x − x′) whose
- dictions with observed exponents is still not completely sign depends on the chirality. For the antidot of length
d
n understood, although severalmechanisms have been put L, the field φA(x) comprises a zero-mode describing the
o forward,includingcouplingtophonons[7,8],interaction charged excitations and a neutral boson satisfying pe-
c with reservoirs[9], and edge reconstructionwith smooth riodic boundary conditions, HA0 = Ecn2 + l>0lǫa†lal.
v: confining potentials [10]. Here,Ec =πνv/ListhetopologicalchargeexPcitationen-
i ergy,andn is the excessnumber ofelementaryquasipar-
X In this Letter, we aim to find signatures of fractional ticles; for the neutral sector, a,a† are bosonic operators
r charge in different transport regimes, and to distinguish (plasmons) and ǫ = 2πv/L is the plasmonic excitation
a
them from effects due to quasiparticle propagators. We energy [12]. The effect of HAB is merely to shift the en-
consider a system consisting in a quantum Hall sample ergies in HA0 according to Ecn2 → Ec(n −ϕ)2, where
with an embedded antidot (Fig. 1(a)) at filling factor ϕ = Φ/Φ0 is the Aharonov-Bohm flux Φ through the
ν =1/p. We derive unambiguous signatures of the frac- antidot measured in flux quanta Φ0 =hc/e [12].
tionalchargein processeswithdifferent transportstatis- The term HT = t ψ†(x )ψ (0) + h.c. repre-
i=L,R i A i i
tics through a comparison of noise and skewness in the sents the most relevaPnt processes of single-quasiparticle
sequentialregime. Inaddition,wefindtransportregimes tunneling [3, 12]. Here, ψ†(x)∝e−iφl(x) are the creation
l
wheretheFanofactorissensitivetothepowerlawsofthe operators for quasiparticles in the leads and in the an-
quasiparticle propagators and presents super-poissonian tidot. Standard commutation relations ensure a charge
2
ϕ ments as a tool to determine the χLL exponent [9] and
(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)((cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)a(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1))(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)L0txLL (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)A(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)xRtR0R(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (nnb++n)12(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)0(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)I(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)E(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)E+In(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)2(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)n−I I4II I6V e∗VV/Ec tmchoaentsiciodanrerroitenhrtechhnea-strthgaeot,irsddkteeincrcson=uoofprtlm|ihenahenghl−Iitztr1iehahidnIneiscl|pua.otrtrreetrnptfrrcooucmemsust.lhaWentien[w1f(o63irl])-l,
Here, hIi is the stationary current and hhIii =
n
FIG. 1: (a) Geometry of the system. (b) Scheme of trans- limτ→∞(e∗)nhhNτiin/τ is the n-th irreducible current
portregionsinthe(V,ϕ)plane. Romannumbersindicatethe moment given in terms of the irreducible moments of
number of charge states involved in the transport - hatched, the number N of charge e∗ particles transmitted in the
τ
blockade regions. Thin lines signal the onset of transitions, timeτ. Fanofactorandnormalizedskewnesscorrespond
whereenergiesE±n =0(seetext). Thicklinesindicatethedi- to k .
amond where plasmonic excitations first enter into effect for 2,3
The statistics of a transport process is completely iden-
ν =1/3 (region V).
tifiedbyitscumulantshhN ii . Indeed,ifaprocesswith
τ n
a given statistics takes place at different filling factors
of quasiparticles e∗ = νe [2]. The tunneling probability with e∗1 = ν1e and e∗2 = ν2e, then the comparison of
ratiobetweenthetwobarriersistunedbyanasymmetry the n-th order current cumulants gives direct informa-
η = |tR|2/|tL|2. A source-drain voltage V is applied be- tion on the charge ratio according to kn(ν1)/kn(ν2) =
tweentheleftandrightedges,producingabackscattering (e∗1/e∗2)n−1 [17]. We suggest to revert this approach to
tunneling current of quasiparticles through the antidot detect the chargefractionalizationin our antidotgeome-
I(t)=[I (t)−I (t)]/2, with (j =L,R) try. To do so, we define special the conditions in the pa-
L R
rameter space where our system has the same transport
I (t)=ie∗ t ψ†(x ,t)ψ (0,t)−h.c. . (2) statistics for different filling factors and independently
j (cid:20) j A j j (cid:21) from the value of g [18]. Note that a comparison of all
moments would be required to identify special regimes.
Sequential tunneling rates - For sufficiently small tun-
Here, we will adopt only the minimal comparison of the
neling as compared with temperature, transport can
secondandthird momentthatare moreaccessiblein ex-
be safely described within the sequential tunneling
periments. Furthermore, unlike simpler geometries our
regime[13]. Here,themainingredientsaretheincoherent
system offers the possibility to identify several special
tunneling rates Γ (E). Their expressionis well known
L,R points with different statistics by changing external pa-
withinthe Luttingerdescriptionofedgestateswithfully
rameters.
relaxedplasmonicexcitationsoftheantidot[14,15]. One
Thedetailedanalysisofk isobtaineddirectlyfromthe
has Γ (E) =|t |2Γ(E) = |t |2 w γ(E−lǫ) where E is 2,3
i i i l l cumulant generating function calculated in the marko-
theenergyassociatedtothequPasiparticletunnelingevent
vian master equation framework [19] in the sequential
and γ(x) = (βω /2π)1−g|Γ(g/2 + iβx/2π)|2eβx/2 with
c regime. The stationary occupation probability of a fixed
Γ(x) the Euler Gamma function and β = 1/k T. The
B number of antidot quasiparticles is obtained in analogy
factors w are function of the plasmonic energy ǫ, the
l to the electronnumber occupationinquantumdots [14].
interaction parameter g and the cut-off energy ω [15].
c Assuming a symmetric voltage drop at the barriers, the
Note that in the standard χLL theory g = ν. Here, we
change in energy for the forward transitions n → n+1,
will assume g = νF in order to describe possible renor-
n+1 → n is En = e∗V/2±2E (ϕ−n−1/2) respec-
malization effects due to coupling of the infinite edges ± c
tively. The conditions En =0 grid the (V,ϕ) plane into
with additional modes, e.g. phonons, or to edge recon- ±
diamonds according to the scheme in Fig. 1(b).
struction. The explicit value of F will depend on the
Results - We focus at first on the few-state regime
details of interaction [7, 8, 9, 10] and here we will con-
e∗V . 2E . In regions I transport is suppressed; lin-
c
sideritasafreeparameter. Notealsothatthefractional
ear conductance oscillations exist in regions I,II with a
chargee∗ issolelydeterminedbyν andisthus separated
periodicityofafluxquantumΦ foranyν,inaccordance
0
from the dynamical behaviour governedby g.
with gauge invariance [20]. In the same regime, an an-
For g < 1 the rates scale at low temperatures as Tg−1
alytical treatment of k is possible. Since the energy
2,3
at energy E = lǫ. This behaviour is reflected in the in-
spectrum is periodic in ϕ, we start at n = 0 . Here the
creaseofthe linearconductancemaximumG ∝Tg−2
max forward tunneling rates Γ0 = Γ E0 dominate the dy-
with decreasing temperature. In order to be consis- ± ±
namics and we recover a known(cid:0)form(cid:1)ula [15, 21] for the
tent with the tunneling approximation we then require
Fano factor
G ≪ e2/h, setting a limit to the low temperature
max
regime [13]. k2 =coth(βe∗V )−2ηΓ0+Γ0−f−(e∗V), (4)
Moments -Hereafter,wewillstudyhighercurrentmo- ν 2 Γ2
tot
3
0.5 5
k 1 1 k
2 3 ϕ k
ν 0.8 0.8 ν2 ν2
0
0.6 0.6
-0.5
0.4 0.4
0.5
0.52
0.2 0.2 ϕ
0.253
0
0 0.50 0.250 0 1
(a) 0.49 0.50 (b) 0.49 0.50
-0.5 0
0.48 0.50 0.52 0.48 0.50 0.52
ϕ 0 2 4 6 e∗V/E
c
FIG. 2: Shot noise limit with η = 1.5, e∗V = 0.1E , k T =
c B FIG. 3: Fano factor k /ν at ν =g =1/3, k T =0.01E , vs.
2 B c
0.004E and different g =1/5 (solid), g =1/3 (dotted), g =
c source-drain voltage and magnetic flux. Top panel: symmet-
1/2 (dashed). (a) Fano factor k /ν and (b) skewness k /ν2
2 3 ric barriers η =1. Bottom panel: strong asymmetry η =10.
vs. magnetic flux ϕ. Insets: zoomed regions of the minima,
Right panel: color scale.
with grey lines at (a) 1/2 and (b) 1/4.
tics (strongest anticorrelation) for any g ≤1.
where Γ = Γ0f (E0) + ηΓ0f (E0) with f (x) =
tot + + + − + − ± Intheintermediateregimee∗V ≈k T,k dependmore
1±e−βx. B 2,3
stronglyontheparameterg,andtheinterplayoftwoen-
For the skewness we find
ergy scales prevents the onset of special regimes.
k3 =1−6ηΓ0+Γ0−f+(e∗V)+12η2Γ0+2Γ0−2f−2(e∗V). (5) facTtohrerims ainldreepgeimndeene∗tVfro≪mktBhTe.chIanrgtehifsralcimtioitnathliezaFtiaonno,
ν2 Γ2 Γ4
tot tot k =2k T/eV,reflectingthefluctuation-dissipationthe-
2 B
Weanalyzenowthebehaviours(4)and(5)varyingthe orem. This is not true for the normalized skewness,that
ratio e∗V/k T. measuresthe fluctuation asymmetryinducedby the cur-
B
Shotnoiselimitk T ≪e∗V. IntheblockaderegionsI rent. In this regime, the skewness opens the possibility
B
with|βE0|≫1,onehask =νandk =ν2. Inthiscase to measure the carrier charge e∗ = νe that is no more
± 2 3
the statistics of the transport process is poissonian: the addressableviathe Fano factor. Indeed, forlow voltages
transport through the antidot is almost completely sup- V →0+ one has
pressed,I ≈0,andtheresidualcurrentisgeneratedonly
η 1
byathermallyactivatedtunnelingthatiscompletelyun- k =ν2 1−3 , (6)
correlated. SoinregionIforafixedvalueoffillingfactor, 3 (cid:20) (1+η)2Cosh2(βEc(ϕ−1/2))(cid:21)
k take maximal values corresponding to a poissonian
2,3 that does depend on ν but not on the exponent g.
transport process, thus constituting an example of spe-
We study now higher voltages e∗V > 2E where the
cial regime. c
renormalized interaction parameter g has a prominent
We consider the two-state regime (II) for βE0 ≫1. For
± role. For this purpose we consider the behaviour of the
fractional edges g < 1, k have a particular functional
2,3
Fanofactor. Hereingeneralanumericalapproachisnec-
dependenceonϕ. Wefindthattheybothdevelopamin-
essary. In Fig. 3 a density plot of k for ν = g = 1/3 as
imum [22] and that the absolute values of the minima 2
a function of magnetic field and source-drain voltage is
are, respectively, kmin = ν/2 and kmin = ν2/4. These
2 3 shown for different asymmetries. We recover that, in-
minimal values do not depend on g, as the compari-
dependently from η, in region I one has k = ν that
son of solid (g = 1/5), dotted (g = 1/3) and dashed 2
corresponds to a poissonian statistics. We will thus re-
(g = 1/2) curves in Fig. 2 confirms. For Fermi liq-
fer to the red (blue) regions, where k > ν (k < ν), as
uid edges g = 1, we have k = ν(1+η2)/(1+η)2 and 2 2
2
super(sub)-poissonian noise regimes.
k =ν2 1−6η(1+η2)/(1+η)4 independently from ϕ.
3
Inthethree-stateregimeIII,acomparisonofthetopand
Here, k2(cid:2) and k3 assume their m(cid:3)inimal values ν/2 and
bottompanelsshowsthatsuper-poissonianvaluesarein-
ν2/4 in the symmetric case η =1. In this conditions we
duced by high barrier asymmetry. The Fano factor in
have the strongest anticorrelation that is signalled by a
thisregimedepends onalargersetofratesΓn =Γ(En),
marked sub-poissonian statistics. ± ±
n
n = 0,1, and the corresponding backward rates Γ =
We can conclude that in the two-state regime, in the ±
shot noise limit, the values of the minima for k2,3 ob- e−βE±nΓn±. A tractable analyticalformula canbe derived
tained varying η,ϕ correspond to a special condition underthereasonableassumptionthatonlytwobackward
0 1
where the system shows the same sub-poissonian statis- rates, Γ and Γ , survive: one has k /ν = 1−2η δk ,
− + 2 2
4
especially on account of recent accomplishments in mea-
1.1
k2 surement techniques applied to electron counting [24].
ν 1
0.9
Financial support by the EU via Contract No.
0.8 MCRTN-CT2003-504574is gratefully acknowledged.
0.7
0.6
1.5 2 2.5 3 3.5 [1] R. B. Laughlin, Phys.Rev.Lett. 50, 1395 (1983).
e∗V/E [2] X.G.Wen,Phys.Rev.Lett.64,2206(1990);Phys.Rev.
c
B 41, 12838 (1990); 43, 11025 (1991).
[3] C.L.KaneandM.P.A.Fisher,Phys.Rev.Lett.72,724
FIG.4: Fanofactork /ν vs. e∗V/E atϕ=0,k T =0.05E .
2 c B c (1994).
Color code: red g = 1/3, green g = 0.8, black g = 1. Solid
[4] R.de-Picciotto,M.Reznikov,M.Heiblum,V.Umansky,
lines η=1, dashed lines η=10. Grey line, poissonian limit. G. Bunin, and D. Mahalu, Nature (London) 389, 162
(1997); L. Saminadayar, D. C. Glattli, Y. Jin, and B.
Etienne, Phys. Rev.Lett. 79, 2526 (1997).
[5] A.M.Chang,L.N.Pfeiffer,andK.W.West,Phys.Rev.
with Lett. 77, 2538 (1996); M. Grayson, D. C. Tsui, L. N.
Pfeiffer, K. W. West, and A. M. Chang, ibid. 80, 1062
δk =Γ0t2Γ1+Γ1−+Γ1t2Γ0+Γ0−+Γ0−Γ1+ Γ0t−Γ1t ηΓ1−−Γ0+ . (1998).
2 ηΓ0Γ1+Γ0 Γ1(cid:0)+Γ1 2(cid:1)(cid:0) (cid:1) [6] S.Roddaro,V.Pellegrini, F.Beltram, G.Biasiol, andL.
− t t + t Sorba, Phys.Rev.Lett. 93, 046801 (2004).
(cid:2) (cid:0) (cid:1)(cid:3) (7) [7] B. Rosenow and B. I. Halperin, Phys. Rev. Lett. 88,
Here, Γ0 = Γ0 +ηΓ0 and Γ1 = ηΓ1 + Γ1. We note 096404 (2002).
t + − t − +
that in order to have super-poissonian noise a fractional [8] O. Heinonen and S. Eggert, Phys. Rev. Lett. 77, 358
g < 1 is necessary, with additional conditions on the (1996).
[9] V. V. Ponomarenko and N. Nagaosa, Phys. Rev. B 60,
asymmetry. Indeed, setting η =1 in Eq. (7) in the limit
16865 (1999).
βE1,βE0 ≫ 1 yields δk > 0 for any g. On the other
+ − 2 [10] K. Yang, Phys.Rev.Lett. 91, 036802 (2003).
side, setting g =1 gives δk =2η/(η2+η+1)2 >0.
2 [11] V.J.Goldman,JunLiu,andA.Zaslavsky,Phys.Rev.B
Soitappearsthatpositivecorrelationsareinducedbyan 71,153303(2005);V.J.Goldman,I.Karakurt,JunLiu,
interplay of η and g. Figure 4 shows the Fano factor as and A.Zaslavsky, ibid. 64, 085319 (2001).
a function of e∗V/E for different η and g. Here ϕ = 0, [12] M.R.GellerandD.Loss,Phys.Rev.B56,9692(1997).
c
althoughsimilarconsiderationsapplyingeneral. Forg = [13] A. Furusaki, Phys.Rev.B 57, 7141 (1998).
[14] M. R. Geller and D. Loss, Phys. Rev. B 62, R16298
1,k remains sub-poissonian(blacklines), while positive
2
(2000).
correlations appear for g < 1 and sufficient asymmetry
[15] A. Braggio, R. Fazio, and M. Sassetti, Phys. Rev.B 67,
(color lines).
233308 (2003).
Finally, interesting effects take place in the five-state [16] L.S.LevitovandM.Reznikov,Phys.Rev.B70,115305
regime (V) for ν = 1/3. Here, a superpoissonian Fano (2004).
factor appears along the diamond lines for η = 1 (see [17] H. Saleur and U. Weiss, Phys. Rev. B 63, 201302(R)
Fig. 3 top) and disappears for large asymmetries (Fig. 3 (2001).
[18] In the point contact geometry the tunneling limit is a
bottom). Detailed investigations [23] show that the col-
special regime, with Poissonian statistics e.g. for both
lective excitations of the antidot edge are responsible of
ν =1 and ν =1/3.
thesuper-poissonianbehaviouratsmallasymmetries. In [19] D. A. Bagrets and Yu. V. Nazarov, Phys. Rev. B 67,
this region, in fact, the tunneling process can excite the 085316 (2003).
plasmonicmodesofenergyǫ=2Ec/ν thatexactlycorre- [20] N.Byersand C. N.Yang,Phys.Rev.Lett. 7, 46(1961).
spond to the diamond lines (Fig. 1(b), thick curves). In [21] Y.M.BlanterandM.Bu¨ttiker,Phys.Rep.336,1(2000).
particular, one can show [23] that k shows a superpois- [22] An analytical approximation for the minimum is given
2
sonian maximum as a function of e∗V with a peculiar by ϕmin =1/2−(e∗V/4Ec)(1−ηeαff(g))/(1+ηeαff(g)), with
scalinglawkmax ∝Tg−1 directlyconnectedtotherenor- ηeff =η[1+O(βEc)].
2 [23] A. Braggio, M. Merlo, N. Magnoli, and M. Sassetti (un-
malized lead exponent.
published).
In conclusion, we have found distinct, unambiguous
[24] Yu.Bomze, G. Gershon, D.Shovkun,L. S.Levitov,and
signatures of fractionalcharge and interactionrenormal- M.Reznikov,Phys.Rev.Lett.95,176601(2005);S.Gus-
izationinhighmomentsoftunnelingcurrentinapromis- tavsson, R. Leturcq, B. Simovic, R. Schleser, T. Ihn, P.
ing Hall-antidot geometry [11]. Confirmation of such Studerus,K.Ensslin, D.C. Driscoll, and A.C.Gossard,
novel results appears to be within experimental reach, cond-mat/0510269.