Interacting electron systems between Fermi leads: effective one-body transmissions and correlation clouds

0.19 MB·English

by  Rafael A. Molina

EPJ manuscript No.
(will be inserted by the editor)
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Interacting electron systems between Fermi leads:
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2 effective one-body transmissions and correlation clouds

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J Rafael A. Molina1,2, Dietmar Weinmann3, and Jean-Louis Pichard1,4

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2 1 CEA/DSM, Service dePhysiquede l’Etat Condens´e, Centred’Etudes deSaclay, 91191 Gif-sur-Yvette,France
2 Max-Planck-Institutfu¨r Physik Komplexer Systeme,N¨othnitzer Str.38, 01187 Dresden,Germany
] 3 Institut de Physique et Chimie des Mat´eriaux de Strasbourg, UMR 7504 (CNRS-ULP), 23 rue du Loess, BP 43, 67034
l
e Strasbourg Cedex 2, France
- 4 Laboratoire de PhysiqueTh´eorique et Mod´elisation, Universit´edeCergy-Pontoise, 95031 Cergy-Pontoise Cedex, France
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Reference: Eur. Phys.J. B 48, 243-247 (2005)
.
t
a
m Abstract. In order to extend the Landauer formulation of quantum transport to correlated fermions, we
consideraspinlesssysteminwhichchargecarriersinteract,connectedtotworeservoirsbynon-interacting
-
d one-dimensionalleads.Weshowthatthemappingoftheembeddedmany-bodyscattererontoaneffective
n one-body scatterer with interaction-dependent parameters requires to include parts of the attached leads
o where the interacting region induces power law correlations. Physically, this gives a dependence of the
c conductanceofamesoscopicscattereruponthenatureoftheusedleadswhichisduetoelectroninteractions
[
inside the scatterer. To show this, we consider two identical correlated systems connected by a non-

2 interacting lead of length LC. We demonstrate that the effective one-body transmission of the ensemble
v deviates by an amount A/LC from the behavior obtained assuming an effective one-body description for
1 eachelementandthecombinationlawofscatterersinseries.AismaximumfortheinteractionstrengthU
7 aroundwhichtheLuttingerliquidbecomesaMott insulatorintheusedmodel,andvanisheswhenU →0
5 and U →∞. Analogies with theKondo problem are pointed out.
7
0 PACS. 71.27.+a Strongly correlated electron systems; heavy fermions – 72.10.-d Theory of electronic
5 transport; scattering mechanisms – 73.23.-b Electronic transport in mesoscopic systems
0
/
t
a 1 Introduction in molecules [3], atomic chains or contacts [4], quantum
m
dots where few electrons might form a correlated solid, a
- In Landauer’s formulation of quantum transport [1], the chargedensity wave,a Mottinsulator,etc.Inthese cases,
d
n measure of the conductance g of a coherent system is the electrons are transmitted from one Fermi reservoir to
o formulated as a scattering problem between incoherent another through a many-body scatterer. To extend Lan-
c electron reservoirs. In a two-probe geometry, the system dauer’s approach to such systems, at least for low tem-
: is connected to two reservoirs via leads. For large elec- peratures and bias voltages,one needs to reduce the bare
v
i tron densities, the Coulomb interaction is screened and many-body scatterer to an effective one-body scatterer
X the Coulomb to kinetic energy ratio rs is small. One has with interaction-dependent parameters. This task will be
r essentially a non-interacting system of Fermi energy EF, hopeless for an isolated system where electrons interact
a
where the occupation of the one-body levels is given by with a largeinteractionstrength U, but becomes possible
a Fermi-Dirac distribution at a temperature T. The sys- when leads where electrons do not interact are attached
tem acts as a one-body scatterer and its residual conduc- to it. This has been numerically demonstrated in previ-
tanceg(T →0)is given(in units of2e2/hforsinglechan- ous works [5,6] using the embedding method, which al-
nelleadsandspindegeneracy)bytheprobability|t(EF)|2 lows to extract [5,6,7,8,9,10,11] the effective coefficient
of an electron of energy EF to be elastically transmitted |t(EF,U)|2 from the persistent current of a large non-
through it. interacting ring embedding the many-body scatterer. Us-
The problem of describing coherent electronic trans- ing the same method, we show that it is not the region
port becomes more complex in the case where the car- where the electrons interact which acts as an effective
rier density is low inside the scatterer,the screeningceas- one-body scatterer with renormalized parameters, but a
ing to be effective and the electrons becoming correlated. larger region where the many-body scatterer induces cor-
Suchsituationsoccurinquantumpointcontactsoftrans- relations.This problemissomewhatsimilarto the Kondo
verse size smaller than the Fermi wavelength, where a problem, which can be solved using Wilson’s numerical
0.7 (2e2/h)structure is observed[2], andcanbe expected renormalization group (NRG) [12]. Instead of using the
2 R.A.Molina et al.: Correlation clouds around interacting electron systems
LC
LS LS
φ
Fig. 2. Scheme of the set-up with two identical many-body
scatterers connected by LC sites where the carriers do not in-
LS teract. When LC is small, thetwo correlation clouds sketched
in grey overlap and the effective one-body scatterer of theen-
sembleisnot given bytheeffectiveone-bodyscatterer ofeach
Fig. 1. Scheme of the ring pierced by a flux φ used for the
elementandthecombination lawofone-bodyscatterersin se-
embedding method. The correlation cloud induced by LS in-
ries.
teracting sites upon theauxiliary lead is sketched in grey.
The scattering geometry corresponds to two leads of
NRG method, we use the density matrix renormalization
group (DMRG) method [13,14] for a non-interacting ring LL/2 → ∞ sites connected by an interacting scatterer
embeddingthemany-bodyscatterer,assketchedinFig.1. of LS sites. The electrons do not interact in the leads,
a necessary condition for having appropriate asymptotic
The modulus of the effective one-body transmission am-
scattering channels in one dimension. For the embedding
plitude |t(EF,U)| is obtained from the persistent current
method,weconsidertheringgeometrysketchedinFig.1,
of the ring extrapolated to infinite lead length, while its
the scatterer being closed on itself via a non-interacting
phase α is given by the Friedel sum rule.
After a study of the contained extrapolation, we ap- leadofLLsites.Thisisachievedbyaddingahoppingterm
ply the embedding method to determine the effective
total transmission coefficient |tT(EF,U)|2 of two identi- −thc†1cLexp(iφ)+h.c.
cal many-body scatterers in series, connected by a non-
to the Hamiltonian (1), the flux φ driving a persistent
interacting lead of size LC (sketched in Fig. 2). The re-
currentJ(U) in the ring. As the flux dependence of J(U)
osublttainingedexaascstuvmailnugetfohre|ctoTm(EbFin,aUti)o|2ndlaewviaotfeosnfer-obmodtyhescoante- extrapolated to the limit LL → ∞ demonstrates [6], the
many-body scatterer behaves as an effective one-body
terers in series. This U-dependent deviation is due to in-
scatterer,butwithaninteraction-dependentelastictrans-
duced correlations in the attached leads, and its depen-
dwehnicche aocntsLaCs aanlloewffsecttoivdeeotnerem-biondeytshceatstiezreero.f the region smimispsiloenr [c6o]etffiocgieentt|t|(t(EEFF,,UU))|2|2f.rIonmstethade cohfaursginegstJiff(Une)s,sitis
L
D(U,LS,L)=(−1)N E0(U,LS,L)−Eπ(U,LS,L) ,
2
2 Embedding method, extrapolation and (cid:0) (cid:1)(2)
correlation cloud where E0(U,LS,L) − Eπ(U,LS,L) is the change of the
ground-state energy from periodic to antiperiodic bound-
Tostudythemappingofabaremany-bodyscatterercou- aryconditions.D(U,LS,L)isobtainedbytheDMRGim-
pled to leads onto an effective one-body scatterer with plementation for real Hamiltonians, which can be used to
interaction-dependent coefficients, we take a model of N study with a great accuracy systems as large as L = 120
spinless fermions in a chain of L = LS +LL sites. The sites with N = 60 particles. In the limit LL → ∞, one
Hamiltonian (with even LL) reads gets the modulus
H =−thXi=L2(c†ici−1+c†i−1ci) |t(EF,U)|=sin(cid:18)π2D∞D(∞U(U=,L0,SL)S)(cid:19) (3)
(1)
LL/2+LS ofthe transmissionamplitude throughthe scattererofLS
+U [ni−V+][ni−1−V+] . sites, D∞(U = 0,LS) being the charge stiffness of the
i=LXL/2+2 same ring for U =0.
To take the limit LL → ∞ is one of the key points
The hopping amplitude th = 1 between nearest neigh- of the embedding method. This extrapolation is also re-
bor sites sets the energy scale, c (c†) is the annihilation
i i quired for pure one-body scattering, where the finite size
(creation) operator at site i, and ni = c†ici. The near- corrections to formula (3) can be expanded [6] in powers
est neighbor repulsion U acts upon LS consecutive sites of1/L.Formany-bodyscattering,theDMRGstudygives
and gives rise to many-body scattering. We take a half- an empirical scaling law [5,6]
filled model (N = L/2), with a potential V+ = 1/2 being
due to a positive background charge which exactly com- C(U,LS)
pensates the repulsion U inside the scatterer. Therefore, D(U,LS,L)=D∞(U,LS)exp(cid:18) L (cid:19) (4)
our model exhibits particle-hole symmetry and a uniform
density,without Friedel oscillationsaroundthe scattering obtainedforlargeLL andsmallLS,whichallowstodeter-
region where the fermions interact. minetheasymptoticvalueD∞(U,LS)necessarytoobtain
R.A.Molina et al.: Correlation clouds around interacting electron systems 3
2 1
|t(EF,U)| . Expanding this scaling law gives
B(U,LS)
D(U,LS,L)−D∞(U,LS)≈ L (5) 0.9
when L is large enough, where
g
B(U,LS)=C(U,LS)D∞(U,LS). (6) 0.8
This is a power law decay, and not an exponential de-
cay with a characteristic scale above which the finite 0.7
size correction can be neglected. Numerical data show
2 4 6 8 10 12 14 16 18
that B(U,LS) is important for intermediate interaction
strengths U. But in the limits U → 0 (no scatter- L
C
ing, total transmission) and U → ∞ (total reflection,
Fig. 3. Conductance g(LC) of the set-up sketched in Fig. 2
D∞(U → ∞,LS) → 0), the finite size corrections vanish with LS = 2, for U = 1. The points are obtained with the
and B(U,LS)→0. embedding method for the ensemble. The dashed line at g =
However, since adding one-body potentials in the re- 0.717givestheapproximatevalueobtainedfromEq.(15),with
gion of the LS sites yields a finite size correction to |t| obtained by the embedding method for a single scatterer.
D(U,LS,L)even when U →0, the interpretationof these Thesolidlineisafitwiththeformg(LC)=0.7174−0.057/LC.
corrections is not straightforward. They do not depend
only on the correlations induced in the attached lead by 1
the interaction acting inside the scatterer, but also on 0.99
more trivial one-body aspects.
0.36
3 Combination of two many-body scatterers
g
in series
0.34
A more direct approach, where the finite size effects are
onlyduetomany-bodycorrelations,consistsintakingtwo 0.32
identical scatterers connected by a scattering-free lead of
sizeLC inwhichtheelectronsdonotinteract,assketched 2 4 6 8 10 12 14 16
in Fig. 2. Since the scattering channels begin at the first L
C
attached sites of the leads when U = 0, there is a sim-
Fig. 4. Conductance g(LC) as in Fig. 3, but for U = 2. The
ple combination law for one-body scatterers in series. To
dashed line at g = 0.357 represents the value yielded by Eq.
study how this combination law is brokenwith increasing
(15), and the solid line is thefit g(LC)=0.3572−0.077/LC.
U when LC is small allows to show that the size of the
effective elastic scatterer is larger than the region where
the carriers interact. When U 6= 0, the scattering chan- be written as ψi0 = cos(ki+δ0) and ψi1 = sin(ki+δ1) at
nels begin only asymptotically far from the many-body the right side of the scatterer, and ψi0 =cos(ki−δ0) and
scatterer. ψi1 = sin(ki−δ1) at its left side. The two phase shifts δ0
Without interaction, a scatterer can be described at
and δ1 are related [15] to t and r by
energy EF by a unitary scattering matrix SS, written in
terms of its reflection and transmission amplitudes r,r′ t=(exp(2iδ0)+exp(2iδ1))/2,
and t,t′ as (9)
r t′ r=(exp(2iδ0)−exp(2iδ1))/2.
SS =(cid:18)t r′(cid:19) . (7)
Due to symmetries, SS or MS have only two free pa-
The scatterer being symmetric upon time reversal, one rameters: the modulus |t| = cos(δ0 −δ1) and the phase
has t=t′, while r =r′ if the scatterer is symmetric upon α = δ0+δ1 of the transmission amplitude t, the unitar-
space inversion. The transfer matrix MS (giving the flux ity of SS (|t|2+|r|2 = 1 and r/r∗ = −t/t∗) giving r. We
amplitudesattherightsideintermsofthefluxamplitudes can determine |t| by the embedding method. The Friedel
at the left side) reads sum rule [16] gives α. If one introduces a scatterer with
inversion symmetry in the central region of a scattering
1/t r/t free lead, this rule states [17] that
MS =(cid:18)r∗/t∗ 1/t∗(cid:19) . (8)
α=δ0+δ1 =πNf (10)
The total Hamiltonian and the parity operator can be si-
multaneously diagonalizedif one has inversionsymmetry, for spinless fermions in one dimension. Nf is the number
to give even and odd standing-wave solutions which can of displaced fermions when the scatterer is introduced in
4 R.A.Molina et al.: Correlation clouds around interacting electron systems
0.1
U=2 LS=2
U=1 LS=4
U=4 0.1 LS=6
g L)S
δ0.01 U ,
A (0.05
0.001 0
1 2 4 8 16 0 2 4 6 8
L U
C
Fig. 5. Errorδg madewhenusingthecombinationlawofEq. Fig. 6. Amplitude A(U,LS) of the fits shown in Fig. 5 as a
(14)forhavingtheconductanceoftheset-upsketchedinFig.2 function of U for different values of LS.
withLS =2anddifferentvaluesofU,asafunctionofaneven
numberLC ofconnectingsites.ThelinesgiveanA(U,LS)/LC 4 Correlation-induced deviations from the
fit.
non-interacting combination law
the central region. For a uniform filling factor ν = 1/2,
Fig. 3 and 4 show the conductance g for two scatterers of
Nf =LS/2, and the phase α reads LS = 2 sites in series, as a function of the length LC of
the coupling lead. The data points are directly obtained
α=πNf = πLS =kFLS, (11) from the embedding method, without assuming a combi-
2 nation law for scatterers in series. Resonances with g =1
occurforoddLC ={1,3,5,...}.ForevenLC,the dashed
where kF = π/2 is the Fermi wave number. For the spin- lines represent the LC-independent values |t|4/(|t|2−2)2
less case in one dimension with a uniform density, this
implied by Eq. (14), the coefficient |t| being obtained us-
simply means that the transmitted wave has Nf changes ing the embedding method for a single scatterer. Within
of sign when one transfers a fermion through a scatterer
the accuracy of the extrapolation procedures required for
containing Nf others. This is obvious for U = 0 as well having the transmission |t| of an individual scatterer and
as for U 6= 0. Using the same rule, the ideal ballistic
the total conductance g, the result of (14) gives the cor-
lead of LC sites has a modulus |t(LC)| = 1 and a phase rect value when LC → ∞, but overestimates g for small
α(LC)=kFLC. Its transfer matrix reads even values of LC. The difference
e−ikFLC 0
MC =(cid:18) 0 eikFLC(cid:19) . (12) δg(LC)=g(LC →∞)−g(LC) (16)
isshowninFig.5forevenLC atdifferentvaluesofU.For
Thecombinationlawofone-bodyscatterersinseriesbeing
even LC, δg(LC) decays as a function of LC as
a simple matrix multiplication for the transfer matrices,
thetotaltransfermatrixMT(EF)oftheensembleisgiven A(U,LS)
by δg(LC)≈ , (17)
LC
MT =MS·MC·MS. (13)
Fseomrbthlee,teoxtparletsrsaendsministseiormncsooefffitchieenttra|tnTs|m2 tishsriooungthotfheeaecnh- fwuinthctiaonnaomftphlietuindteerAa(cUti,oLnSs)trwehnigcthhiUs.shTohwisn1i/nLFCigd.e6caaysias
element and of LC, this gives rDe∞mi(nUi,scLeSn)taonfdthoef1t/hLe sdcerceaeynicnhgaraatcltaerrgizeindgisDta(nUce,sLS(l,aLrg)e−r
thanthe Thomas-Fermiscreeninglength)ofthe potential
|t|4
|tT|2 = 2(1−|t|2)(1+cos(2kFLC−2α))+|t|4 . (14) ocifllaatpiooninst,RchKaKrgYebinytenroanc-tiinotnesra.c.t.i)ngineolencetrdoinmse(nFsrioiend.eTl hosis-
suggeststhat the decay couldbe faster for leads ofhigher
Since |t| = 1 when LS is odd [5,18], we consider only dimensions (1/Ld decay in d dimensions). The amplitude
C
ceovnenduvcatlaunecseogf L=S|.tTT|a2k=ing1αif=LCπLisSo/d2dgiavneds the Landauer wAh(Uen,LUS)→→∞0. IwnhtehnisUlat→ter0lim(oint,e-tbhoedsycastctaetrteerrserbse)coamnde
decoupled from the leads, the energy for an electron to
|t|4
enter or to leave a scatterer being ∝ U. A(U) is maxi-
g = (15)
(|t|2−2)2 mum near U = 2, a value where in the thermodynamic
limit LS → ∞ the Luttinger liquid becomes [19] a Mott
for LC even. insulator for spinless fermions.
R.A.Molina et al.: Correlation clouds around interacting electron systems 5
For all odd values of LC, the data for the total trans- inelasticscatteringvanish[21]whenT →0,inaperturba-
mission coincide with the value g = 1 obtained from Eq. tive approach to the Kubo conductance of an interacting
(14), and δg(LC) = 0. The even-odd dependence on the region embedded between semi-infinite leads. While this
parity of LC shows that the convergences of the phase α agreeswithourfindingsforLL →∞,itislikelythatthese
andthemodulus|t|oftheeffectivescattererarecharacter- corrections do not vanish when LL is finite, and exhibit
ized by different scales.One has α=πNf acrossthe scat- similar power-law decays as LL increases.
terer,directly on a scale LS, independently ofU, while |t|
reachesitsasymptoticvalueonamuchlargerscale.Thisis
notsurprisingsinceαdependsonthe meandensity,while 6 Acknowledgments
|t| depends on the correlations of its fluctuations. In our
modelwitha compensatingbackgroundcharge,the mean We thank Y. Asada, G.-L. Ingold, R.A. Jalabert, O.
densitydoesnotexhibitFriedeloscillations.Letusunder- Sushkov and G. Vasseur for stimulating discussions, and
line thatthe correlationclouds whichhaveto be included P. Schmitteckert for his DMRG code. R.A. Molina ac-
with the many-body scatterer to form the effective one- knowledges the financial support provided through the
body scatterer must not be confused with the screening EuropeanCommunity’s Human PotentialProgramunder
clouds characterizing the charge density. contract HPRN-CT-2000-00144.
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