Kondo screening cloud in a one dimensional wire: Numerical renormalization group study

0.14 MB·English

by  Laszlo Borda

Kondo screening cloud in a one dimensional wire: Numerical renormalization group
study
La´szl´o Borda
Department of Theoretical Physics and Research Group “Theory of Condensed Matter” of the Hungarian Academy of Sciences,
Budapest University of Technology and Economics, Budafoki u´t 8. H-1521 Budapest, Hungary
(Dated: February 5, 2008)
7
WestudytheKondomodel–amagneticimpuritycoupledtoaonedimensionalwireviaexchange
0
coupling– by using Wilson’s numerical renormalization group (NRG) technique. By applying an
0
approach similar to which was used to compute thetwo impurity problem we managed to improve
2
the bad spatial resolution of the numerical renormalization group method. In this way we have

n calculated the impurity spin - conduction electron spin correlation function which is a measure of
a the Kondo compensation cloud whose existence has been a long standing problem in solid state
J
physics. Wealso present results on the temperaturedependenceof theKondocorrelations.

5
2 PACSnumbers: 72.15.Qm,73.63.Kv

]
l
e Introduction— As being one of the most interesting purity only. Even in case of having spin-polarized STM
- quantum impurity problems, the Kondo effect[1] –when tips, it is very difficult to measure the spin correlations
r
t alocalizedimpurityspininteractswiththeitinerantelec- since the Kondo effect manifests itself in continuous, co-
s
t. trons via spin exchange coupling– has been studied for herentspinflipprocessesatatimescale∼1/TK meaning
a decadesboththeoreticallyandexperimentally[2]andhas that one has to measure at a frequency of tens of giga-
m
recently come to its renaissance when it has been pre- hertzs. The idea of suppressing the impurity spin fluc-
- dicted to appear[3] and observed[4] in quantum dot sys- tuations by applying a local magnetic field is doomed to
d
tems. Even though the Kondo effect seems to be well- fail since a magnetic field which is large enough to sup-
n
o understood now, still, there is a controversial question press the spin fluctuations is necessarily large enough to
c which has not been clarified yet: Whether there exists destroy Kondo correlations as well.
[ an extended “Kondo screening cloud” or not? There were experimental setups proposed in order to

2 IncaseofthesinglechannelKondoproblem–whenthe measure the Kondo cloud in confined systems[6], e.g. in
v localspiniscoupledtoonebandofconductionelectrons– case of a quantum dot attached to one dimensional lead
8
0 the interaction with the electrons lifts the ground state with length shorter than, or comparable with ξK. Such
degeneracy of the isolated spin resulting in an effective kind of a proposal suffers from the facts that it is de-
2
1 “screening” of the impurity spin below a certain energy structive, i.e. the Kondo effect disappears for one di-
1 scale, the so-called Kondo temperature TK. It is ade- mensional leads shorter than ξK and one can still argue
6 quatetoaskthe questionwhatisthetypicallengthscale that the suppression of the Kondo effect is a result of
0
inwhichtheimpurityspingetsscreened. Asasimplees- having the level spacing in the lead δǫ > TK for short
/
t timate,justbycomparingthescalesofthecompetingki- enough leads therefore one does not need the theoretical
a
m netic andbinding energies,oneeasilygetsalengthscale, constructionof the screening cloud to explainthe exper-
the so-called Kondo coherence length, ξ = ~v /k T . imentalfindings. Incaseofanimpurityembeddedintoa
- K F B K
d By substituting the typical Fermi velocity vF in metals, higherdimensionalenvironment,anotherlengthscalelK
n and taking a typical Kondo temperature TK 1K, one emerges[7] by equating ∆(lK)=TK where ∆(lK) stands
co getsξK ∼1µm,whichiscomparablewith,ore∼venlarger for the mean level spacing in a box with size lK. For
thanthetypicaldimensionoftoday’smesoscopicdevices. the case of D 2 that length scale can be substantially
v: However, to measure that screening cloud is highly non- smaller than ξ≥K. In 1D systems, which are in the focus
i
X trivial from experimental point of view. ofthe presentpaper,thetwolengthscalesareessentially
equal.
r Inordertoobservethescreeningcloud,onehastomea-
a
sure correlations betweenthe impurity spinand the con- Despite those challenges, there were proposals pub-
ductionelectronspindensity. Itis verydifficult toimag- lished to observe the Kondo screening cloud. These pro-
inesuchkindofameasurementinbulkmetallicsamples. posalsdealwiththe Knightshift[8], persistentcurrent[9]
Recently, it has become possible to perform scanning or conductance of mesoscopic systems[10].
tunneling microscopy (STM) measurements on metallic Very recently, Hand and his coworkers established a
surfaces addressing a single magnetic impurity and its proportionality between the weight of Kondo resonance
close neighborhood[5]. In those experiments the screen- and the spatial extension of the Kondo correlations in
ing cloud was not observed since the STM tip probes mesoscopic systems [11]. This fact allows another, spec-
the localcharge densityofstateswhichisaffectedbythe troscopic way to observe the Kondo screening cloud.
Kondo correlations in the close neighborhood of the im- Those proposals are very promising even though the ex-
2
x
perimental realizations are yet to come.
In the present paper we are less ambitious since our ~k − 1Λ0 . . .
F
goal is not to propose new ways of observation but to impurity ~k −F 1Λ1/2
extend the powerful and numerically exact NRG[12] to ~k − 1ΛΝ/2
F
be capable to compute spatial correlationsand therefore . . . . . .
make predictions for the outcome of future experiments.
Model and definition of correlation functions— The . . .
Hamiltonian of the Kondo model can be written as
a b
H =JS~~σ(0)+ ε c† c , (1)
k kµ kµ
Xk,µ FIG.1: (a)Sphericalshellsinrealspacedepictingtheextents
of wave functions of states which are represented as on-site
whereS~ istheimpurityspin(S =1/2,locatedattheori- states on theWilson chain. As shown, the NRGmethod has
gin),c† (c )creates(annihilates)aconductionelectron a good spatial resolution around the impurity only. (b) A
kµ kµ straightforward extension of NRG tackles that problem pro-
with momentum k and spin µ, while ~σ(0)=ψ†(0)~σψ(0) viding a good spatial resolution at the impurity site as well
standsfortheelectronspindensityatthe positionofthe as at another, freely chosen point by introducing a second
impurity. Asitismentionedabove,thesystemdescribed electron field.
by Eq.(1) forms a singlet below the Kondo temperature
T D exp( 1/2̺ J), where D is the high energy
cuKto∼ff an0d ̺ is−the d0ensity of stat0es of the conduction [D0Λ−n−1;D0Λ−n[ for positive energies measured from
0 the Fermi energy and D stands for the half bandwidth)
electrons. (For sake of simplicity, we consider a band 0
and keeping one mode per interval only. As a next
with a constant density of states in the following. More-
step, one maps the problem onto a semi-infinite chain
over, we restrict ourselves for 1D electrons because it is
with the impurity at the end by means of a La´nczos-
easier to treat them numerically. When it is adequate,
transformation. Asaconsequenceofthe logarithmicdis-
we will comment the case of higher dimensions as well.)
cretization,thehoppingamplitude fallsoffexponentially
A quantity which measures the spatial extension of the
along the chain allowing us to diagonalize the problem
singlet is the equal-time correlator of the impurity spin
iteratively. As it is shown in Ref.[12] the states repre-
and conduction electron spin density at position x:
sented as on-site states on the Wilson-chain correspond
χ (x)= S~~σ(x) . (2) to extended states in real space with a typical spatial
t=0 t=0
h i extension
Since we know that the ground state of the Hamiltonian 1
Eq.(1)isasinglet,wecaneasilyderiveasumruleforthe rN ΛN/2 , (4)
∼ k
F
equal time correlator at zero temperature:
where N is the site index along the chain. It is obvious
∞
3 that the numerical renormalization group method has a
Z hS~~σ(x)it=0dx=−4 . (3) good spatial resolution at the position of the impurity
0 only. [See Fig.1a.]
To tackle that problem, we introduce two electron
Method— To calculate spatial correlations is a non-
fields, one is located at the impurity site, ψ =ψ(0) and
0
trivialtasksincemostofthemethodsusedtoinvestigate
anotheroneatpointxwherewewanttoevaluatethecor-
the Kondo model are not able to reproduce correlation
relationfunction,ψ =ψ(x).[SeeFig.1b.] (Averysimilar
1
functions. Comparatively only a very little of the the-
procedure was applied in the numerical renormalization
oretical study has been focused on spatial correlations:
group treatment of the two impurity problem[17].) Of
Perturbativecalculationshavebeen performed[13, 14]as
course, these fields are not orthogonal, i.e. they do not
well as Monte Carlo analysis[15]. In the present paper
fulfill the canonical anticommutational relations. As a
we extend NRG to compute the Kondo correlations.
next step, we can introduce the proper linear combina-
Wilson’s NRG –the most useful numerically exact
tions of these fields,
method to obtain correlation functions[16]– suffers from
the very bad spatial resolution away from the impu- ψ =1/√2(ψ ψ ) (5)
± 0 1
rity. This fact is a direct consequence of the cor- ±
nerstone of the method, the logarithmic discretization whicharenowanticommutingbutthecorrespondingden-
of the conduction band. In Wilson’s NRG technique sity of states is modified i.e. it acquires energy de-
one introduces a discretization parameter, Λ and with pendence, e.g. for 1D electrons ̺1D(ε) = ̺ /2[1
± 0 ±
the help of that discretizes the conduction band loga- cos(εx/v )]. (In 2D the density of states reads ̺2D(ε)=
F ±
rithmically by dividing it into intervals (i.e. the nth ̺ /2[1 J (εr/v )] where J is the zeroth Bessel func-
0 0 F 0
±
interval is ] D Λ−n; D Λ−n−1] for negative, and tion,whilein3D̺3D(ε)=̺ /2[1 sin(εr/v )/(εr/v )].
− 0 − 0 ± 0 ± F F
3
0 101
j=0.30
j=0.40
100 j=0.50
-0.05 j=0.60
j=0.30
102 j=0.40
χ(x)t=0 -0.1 χ(x) Tt=0K11110000--0211 envelope ~(x/ξ)-2 ~(x/ξ)jjj===-1000...567000 χ(x) Tt=0K1100−−21 T=0
- −
10-3
-0.15
j=0.30 10-140-2 10-1 100 101 10−3
j=0.40 x/ξ
j=0.50 K
-0.20 1 2 3 4 10−4 T=3TK T=TK T=TK/3
k x/π 10−2 10−1 100 101
F x/ξ
K
FIG. 2: (Color online) The equal time spin-spin correlation
function, χt=0(x) = hS~~σ(x)i as a function of the distance FIG. 3: (Color online) The envelope of χt=0(x) for different
x measured from the impurity for different values of the couplingsanddifferenttemperatures: Anyfinitetemperature
Kondo coupling j = ̺0J. χt=0(x) oscillates as ∼ cos2(kFx). introducesanotherenergyscale, ξT =~vF/kBT at whichthe
As shown in the inset the envelope of the oscillating part envelope function crosses over from an algebraic to an expo-
(i.e.χt=0(x=nπ/kF) where n is a integer) crosses over from nential decay.
∼ x−1 to ∼ x−2 at around the Kondo coherence length, ξK.
The envelope function for different couplings nicely collapse
into one universalcurve.
in the inset). The envelope curve (i.e.χ (x = nπ/k )
t=0 F
where n is an integer) changes its decay from 1/x to
As we see, the oscillations of the even/odd density of 1/x2 at around ξ . Our results supports th∼e heuris-
K
states decay as r−(D−1)/2, where D is the dimension- ∼tic picture of the Kondo ground state: The impurity is
∼
alityoftheproblem.) InthenewbasistheKondoHamil- locked into a singlet with an electron whose wave func-
tonian reads tion is spread out over a distance of ξ . If we test the
K
H = 1JS~(ψ† +ψ†)~σ(ψ +ψ ) correlations on a length scale much shorter than ξK the
2 + − + − impurity appears to be unscreened and a perturbative
analysis can be performed[14] yielding χ 1/x. On
+ dε̺ (ε)εc† c . (6) t=0 ∼
Z i iεµ iεµ the other hand, the impurity is screened for distances
X
i=±µ=↑,↓ x ξ and Nozi`eres’ Fermi liquid theory applies which
K
≫
In this representation ~σ(x) = 1/2[ψ+† −ψ−†]~σ[ψ+ −ψ−] rpartehdeicrtswi∼de1c/rxo2ssdoevcearyreignim1De[f1o9r].whTihcherethiesr,ehioswneoverre,lia-
is representedin a highaccuracy. Inbrief, the main idea
able analytic method of calculating χ (x). From per-
behind the above transformation is that one can get a t=0
turbative aspect this is the region where the logarithmic
very good spatial resolution at the impurity site as well
corrections to 1/x becomes large[14]. Alternatively,
as at another freely chosen position x if one is willing to ∼
followingtheargumentationoftheauthorsofRef.11,this
pay the price of (a) having two electron channels mixed
is the region where the impurity can be described by a
by the interaction, (b) having energy dependent density
fluctuating spin resulting in exponential decay which is
of states and (c) having a separate NRG iteration for
overridenby the 1/x2 decay for even larger distances.
everydifferentxvalues. Thosedifficulties arepossibleto ∼
be handled: The most serious one amongst them is the To plot the universal curve in the inset of Fig.2 we
extensionofNRGtoarbitrarydensityofstateswhichhas have chosen the definition of TK based on the NRG flow
been solved by Bulla et al.[18] oftheenergylevels: WeidentifiedTK astheenergyscale
Results— In this NRG scheme the calculation of the atwhichtheenergyofthefirstexcitedstatereaches80%
equal time correlationfunction is rather simple: χ (x) of its fixed point value. We have taken this definition
t=0
appearstobeastaticthermodynamicquantitywhichcan since the energy spectrum is the most basic and most
beevaluatedwithahighprecision. Theresultsfordiffer- reliable result of any NRG calculation and –as a conse-
ent Kondo couplings are shown in Fig.2: the correlation quence of the universality of the Kondo effect– the value
function oscillates as cos2(kFx). As it is shown in the of TK obtained in this way can differ from that of other
insetofFig.2,theenve∼lopeofthecorrelationfunctionfor definitions up to a constant multiplying factor only.
differentKondocouplingsnicelycollapseintooneuniver- NotethattheresultsshowninFig.2correspondto1D:
sal curve (apart from the points x π/k which show in the case of higher dimensions the correlation function
F
≤
non-universal, coupling-dependent behavior, not plotted tends to zero even faster since the sum rule (Eq.3) fixes
4
its integral. This fact enters the numerical calculation
throughthe r−(D−1)/2 decay of density of states oscil-
∼
lations and makes the calculation of χ (x) challenging
t=0 [1] J. Kondo, Prog. Theor. Phys. 32, 37 (1964).
for dimensions higher than D =1.
[2] For a review see A.C Hewson, The Kondo Problem to
Up to this point we have considered the case of zero Heavy Fermions, Cambridge University Press (1993).
temperature T = 0. However, it is known that the tem- [3] L.I. Glazman and M.E. Raikh, JETP Lett. 47, 452
peratureplaysanessentialroleinKondophysics: Kondo (1988); T.K.NgandP.A.Lee,Phys.Rev.Lett.61,1768
correlationsaredestroyedwhen the temperature reaches (1988).
the Kondo scale T T . It is adequate to ask the ques- [4] D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D.
K
∼ Abusch-Magder, U. Meirav, and M.A. Kastner, Nature
tion how does this fact show up in χ (x)?
t=0 (London)391,156(1998); S.M.Cronenwett,T.H.Oost-
Since χt=0(x) is a static quantity we obtain its com- erkamp,andL.P.Kouwenhoven,Science281,540(1998);
pletetemperaturedependencebynumericalrenormaliza-
J.Schmid,J.Weis,K.Eberl,andK.vonKlitzing,Phys-
tion as the iteration proceeds. In Fig.3 we show the re- ica (Amsterdam) 256B-258B, 182 (1998); W.G. van
sults forT =0,T =T /3,T =T ,andT =3T . As it derWiel, S.DeFranceschi, T. Fujisawa, J.M. Elzerman,
K K K
istransparentfromthecurves,theeffectoffinitetemper- S. Tarucha, and L.P. Kouwenhoven, Science 289, 2105
atureshowsupintheappearanceofanotherlengthscale, (2000).
thethermallengthscale,ξ =~v /k T. Forx<ξ the [5] J. Li, W.-D. Schneider, R. Berndt, and B. Delley, Phys.
T F B T Rev. Lett. 80, 2893 (1998); V. Madhavan, W. Chen, T.
correlation function χ (x) is not much affected while
t=0 Jamneala, M. F. Crommie, and N. S.Wingreen, Science
for x>ξT the correlations are cut off exponentially. 280, 567 (1998); H. C. Manoharan, C. P. Lutz, and D.
Based on those results we can now interpret the role M. Eigler, Nature (London) 403, 512 (2000).
of finite temperature in Kondo screening as follows: The [6] P. S. Cornaglia and C. A. Balseiro, Phys. Rev. B 66,
factthattheimpurityspinisperfectlyscreenedatT =0 115303 (2002); P.S.Cornaglia andC.A.Balseiro, Phys.
is reflected in the sum rule given by Eq.(3). At any fi- Rev. B 66, 174404 (2002).
[7] W.B.Thimm,J.Kroha,andJ.v.Delft,Phys.Rev.Lett.
nite temperaturethe integralisreducedandcanapprox-
82, 2143 (1999).
imately rewritten as 0ξT χt=0(x)dx. When the temper- [8] E. S. Sorensen and I. Affleck, Phys. Rev. B 53, 9153
ature is lower than tRhe Kondo temperature T < TK, (1996); J. P. Boyce and C. P. Slichter, Phys. Rev. Lett.
the correspondingthermal length scale is largerthan ξ 32, 61 (1974); Phys. Rev. B 13, 379 (1976); K. Chen,
K
Since χ (x) has its most weight in the region x < ξ , C. Jayaprakash, and H. R. Krishna-Murthy, Phys. Rev.
t=0 K
the effect of a small temperature is just a small correc- Lett. 58, 929 (1987).
[9] I.AffleckandP.Simon,Phys.Rev.Lett.86,2854(2001);
tion to the perfect screening obtained at T = 0. Given
ibid88,139701(2002);H.P.Eckle,H.Johannesson,and
that in the x>ξK regime χt=0(x)∼x−2, the correction C. A. Stafford, Phys. Rev. Lett. 87, 016602 (2001); ibid
appears to be linear in T. In contrast, for T > TK the 88,139702(2002);P.SimonandI.Affleck,Phys.Rev.B
corresponding ξT is shorter than ξK and the correction 64, 085308 (2001); E. S. Sorensen and I. Affleck, Phys.
to the integral is not small and consequently, we cannot Rev. Lett.94, 086601 (2005).
speakaboutthescreeningofthelocalmomentanymore. [10] P. Simon and I. Affleck, Phys. Rev. Lett. 89, 206602
Conclusions— We have shown that Wilson’s NRG (2002).
[11] T.Hand,J.Kroha,andH.Monien,Phys.Rev.Lett.97,
technique is capable to handle spatial correlations if we
136604 (2006).
apply a straightforward extension. To demonstrate that [12] K.G. Wilson, Rev. Mod. Phys. 47, 773 (1975); H.R.
we have computed the S~~σ(x) correlator for a one
t=0 Krishna-murthy,J.W.Wilkins,andK.G.Wilson Phys.
h i
dimensional Kondo system and shown that the decay of Rev. B 21, 1003 (1980); Phys. Rev.B 21, 1044 (1980).
spin correlations crosses over from x−1 to x−2 at [13] R. H. Bressmann and M. Bailyn, Phys. Rev. 154, 471
around the Kondo coherence length∼, ξ = ~v∼/k T . (1967); M. S. Fullenbaum and D. S. Falk, Phys. Rev.
K F B K
We have shown by calculating the temperature depen- 157, 452 (1967); H. Keiter, Z. Phys. 223, 289 (1969);J.
Gan, J. Phys. Condens. Matter 6, 4547 (1994)
dence that any finite temperature introduces a new en-
[14] V.BarzykinandI.Affleck,Phys.Rev.B57,432(1998).
ergy scale beyond which the Kondo correlations vanish
[15] J.E.Gubernatis,J.E.Hirsch,andD.J.Scalapino,Phys.
exponentially. Our method is –apartfrom the numerical Rev. B 35, 8478 (1987).
challenges– easy to generalize to other impurity mod- [16] T.A.Costi,A.C.Hewson,andV.Zlatic,J.Phys.: Con-
els. A very attractiveexample would be the two channel dens. Matter 6, 2519 (1994).
Kondo model to see how the non-Fermi liquid nature of [17] B. A. Jones, C. M. Varma, and J. W. Wilkins, Phys.
thegroundstateshowsupinthespatialspincorrelations. Rev. Lett. 61, 125 (1988); J. B. Silva, W. L. C. Lima,
W.C.Oliveira, J.L.N.Mello, L.N.Oliveira, andJ.W.
The author acknowledges the fruitful discussions with
Wilkins Phys. Rev.Lett. 76, 275 (1996)
G. Zara´nd, R. Bulla, D. Goldhaber-Gordon, A. Za-
[18] R. Bulla, Th. Pruschke, A.C. Hewson, J. Phys.: Con-
wadowski and O. U´js´aghy. This work was supported dens. Matter 9, 10463 (1997); C. Gonzalez-Buxton and
by EC RTN2-2001-00440“Spintronics”,ProjectsOTKA K. Ingersent, Phys. Rev. B 57, 14254 (1998); K. Chen
D048665, T048782, T046303 and by the J´anos Bolyai and C. Jayaprakash, Phys. Rev.B 52, 14436 (1995).
Scholarship. [19] H. Ishii, J. Low Temp. Phys. 32, 457 (1978).