Coherence in the Two Kondo Impurity Problem

0.19 MB·English

by  Lijun Zhu

Coherence in the Two Kondo Impurity Problem
Lijun Zhu and C. M. Varma
Department of Physics and Astronomy, University of California, Riverside, CA 92521
WeshowthroughaperturbativeandanexactcalculationusingWilson’srenormalizationmethods
that in the problem of two interacting Kondo impurities, on-site potential scattering generates a
quantum-tunnelingbetweenthetwoimpuritiesthroughamarginallyrelevantoperator. Themagni-
7 tudeofthistunnelingV12 dependsonthespin-correlationbetweenthetwoimpurities. Forexchange
0
interactionsbetweenthemomentscomparabletotheKondoenergyTK,V12 istypicallymuchlarger
0
than TK. This implies that the heavy-fermion mass in this interesting range is determined by V12
2
rather than TK. The importance of these results for experiments in coupled Kondo dots is also
n pointed out.
a
J

8 The two Kondo impurity problem has been of inter- calculated. These results are of fundamental importance
1 est because it presents the competition between mutual to the problem of the heavy-fermion state.
interactions of local moments and the Kondo renormal- The interactions between the free electrons and the

] izationofthe moments to singletstates ina soluble way. local moments in the two Kondo impurity problem are
l
e An aspect of the problem which has received no atten- [2, 3]
-
r tion hitherto is the the low-energy transfer Hamiltonian
.st oqfuaqsuia-psia-rptaicrlteicplersobbleetmwefeonrathleatttwicoe,imthpisursiettysstihteess.caInletfhoer Himp = J2+ (cid:16)ψ1†~σψ1+ψ2†~σψ2(cid:17)·(S1+S2)
t
a the heavy-fermion bandwidth. The issue is also directly + Jm ψ†~σψ ψ†~σψ (S S )
m posed in a recent ingenious realizationof the two Kondo 2 (cid:16) 1 1− 2 2(cid:17)· 1− 2
d- impurity problem using coupled quantum dots [1]. + J2− (cid:16)ψ1†~σψ2+ψ2†~σψ1(cid:17)·(S1+S2)
n The two Kondo impurity problem was solved by nu-
co mpaerrtiiccalel-hroenleorsmymalmizeattriiocnmgordoeulpobmtaeitnheoddby(NigRnGor)infgorpoa- + KS1·S2+Xσ V (cid:16)ψ1†σψ1σ+ψ2†σψ2σ(cid:17). (1)
[
2 tceonntfioarlmsaclatfiteerlidngth[2e]o,ryand[3]w, absossuobnsizeaqtuieonnt[4ly] satnuddieodthbeyr Here S1 andS2 representtwoS =1/2Kondoimpurities
sittingattwolatticesites. ψ = ψ ;ψ are
v methods[5]. In the single Kondo impurity problem, po- 1(2) Pk 1(2),k 1(2),k
annihilation operators for conduction electrons in spher-
6
tentialscatteringmerelyshiftstheKondoresonancewith
2 icalwavesabouttheimpurity1and2respectively. ~σ are
respect to the chemical potential in accordance with
4 Pauli matrices. J’s are the exchange coupling between
Friedel sum-rule. The sum of the phase shifts in the
7 thelocalmomentsandtheconductionelectrons. V gives
0 even and the odd parity channels for the two Kondo im- the potential scattering; the maximum magnitude of V
6 purity problem is fixed by the Friedel sum-rule to be π.
deduced from the Anderson Hamiltonian is J /4[7]. It
0 Without potential scattering the phase shift in both the | |
is convenientto choose certain J so that the generated
/ even and the odd parity channels is π/2 when the fixed m
t RKKY interaction is 0 while introducing such an inter-
a point is the Kondo fixed point. Consequently, there is
m action K explicitly[2, 8]. For the NRG calculation, the
no splitting of the even-odd resonances or equivalently,
representation in terms of even (e) and odd (o) parities
- no direct hopping of the quasi-particles between the two
d oftheconductionelectronstatesismoreconvenient. The
impurity sites. In this case there is a quantum critical
n latter are obtained by ψ = (ψ ψ )/√2 while
o point (QCP) separating the Kondo singlet regime of the J =J J . e(o),k 1,k ± 2,k
c impurities from the inter-impurity singlet regime [2]. e,o +± −
Note that in Eq.(1) the only connection between the
:
v Affleck and Ludwig [3] pointed out that the QCP is impurity states is through J . (The direct hopping of
−
i changed to a crossover by the potential scattering only the impurity orbitalsfrom one impurity site to the other
X
whentheexchangescatteringatthelocalmomentstrans- is muchmuch smaller than V, J′s or the single-impurity
r
a fers conduction electron from one impurity to the other. Kondo temperature TK for rare-earth or actinide mo-
In this paper, we first explain the Affleck-Ludwig result ments.) This alone (i.e. with V = 0) is insufficient for
byasimpleperturbativecalculation. Wethenobtainthe quasi-particle hopping between sites 1 and 2. This can
newresultthatinsuchacaseaneven-oddsplittingofthe be motivated through a perturbative calculation by the
resonances is generated which depends on the spin cor- diagrams in Fig.(1).
relations. We then present[6] exactresults of a Wilson’s Thevertexcanbeingeneralseparatedasavectorand
NRG calculation to reinforce and quantify these conclu- a scalar part: Γ12 = ΓS,12δαα′ + ΓV,12~σαα′ S. Here
sionsbyderivingthelowenergyHamiltonian. Thesealso S = S +S . The scalar part, Γ , is an eff·ective po-
1 2 S,12
servetogivethecrossoverscaleoftheQCP.Therelation tential scattering electrons between sites 1 and 2. First
ofthecoherencescaletothebareparametersoftheprob- weconsiderthelimitwhentwoimpuritiesarelockedinto
lem and to the generated spin-correlations is explicitly either a singlet or a triplet states, S =0, or 1. The bare
2
(a) (b) (c) where Λ is the discretization energy scale. The impuri-
k1 2k’ 1 2V2 2 1 2 ties only interact with the first site (fp0σ) through the
Hamiltonian H . From the change of spectra with iter-
0
J− J− J+ J+2 2J− ation, the fixed points and the dimensions of operators
V
about them are determined.
For V =0, two distinct spectra are identified for large
FIG. 1: Vertices diagrams for the two Kondo impurity prob-
iterationswiththetuning ofK [2]. WhenK islargeand
lem. (a) shows a bare vertex of J− while (b) and (c) show
leadingcorrectionstoJ− inparticle-particleandparticle-hole ferromagnetic,thespectracanbefittedwiththeeffective
channels,respectively,whichcontributeapotentialscattering Hamiltonian
V12ψ2†ψ1. Solidanddashedlinesrepresentthepropagatorsof H =H∗ +H , (7)
the conduction electrons and the two impurities with a total N N ir
spin S=S1+S2, respectively. wherethefixed-pointHamiltoniansH∗ ineven/odditer-
N
ationsarethesameasthoseoffree-electronsinodd/even
vertex [cf. Fig. 1(a)] is Γ(0) = J . This is corrected iterations, reflecting the π/2 phase shift in both parity
V,12 −
channels and the formation of Kondo resonances. Both
by Fig. 1(b) and Fig. 1(c) in next leading order. They
resonancesdevelopatthechemicalpotential,asshownin
exactlycancelif the intermediate particle-linein (b) and
hole-line in (c) have the same spectra, i.e. at V = 0, or Fig.(2a). Hir containstermssuchastpf0†pf1p+H.c.which
particle-holesymmetry. For V =0,the generatedvertex vanish at N as Λ−(N−1)/2 reflecting the irrelevant
6 → ∞
(including similardiagramswithψ asthe intermediate nature. (t andotherleadingirrelevantparametersserve
1k p
state) is: to determine T ). When K is large and antiferromag-
K
netic,thetwoimpuritiesformaspin-singlet;theeigenval-
S(S+1)
Γ =ρ2 J J (cV), (2) ues of the actualHamiltoniando not show the even/odd
S,12 + −
2 interchange, reflecting 0 or π (for odd and even parity
where ρ is the density of state ofconduction electrons at channels, respectively) asymptotic phase shifts. Near
the chemical potential, S is the total spin of the two im- the QCP at K 2.4T0, singular fermi-liquid behavior,
c ≈ K
puritiesandcisadimensionlessconstantofO(1). Sucha studied in detail earlier[2, 3] ensues.
term in two Kondo impurity problem generates the new
effect, a splitting between even and odd resonances,
V12ψ1†ψ2+h.c.=V12(ψe†ψe−ψo†ψo). (3) (Va=)0 (V bf)inite
V12 also depends on the spin-states of the two impuri- J-=0
odd
ttihees;etffheicstiivsesepeontefnrtoimal tshcaetftaecritntghiantclVu1d2in=gVFeig−. V1(obw-ch)ilies TKTKeven
V ψ†ψ +V ψ†ψ
e e e o o o
S(S+1)
Ve,o =V − 4 cVρ2(J+2 +J−2 ±2J+J−) (4) (V cf)inite (V df)inite
When K , S =1; then J- finite J- finite
→−∞ K < Kc K > Kc
V V(ρJ )(ρJ ). (5)
12 − +
∝
However, when K , S = 0, the contribution to
→ ∞
the splitting due to V vanishes. In between these limits,
there are mixing terms between the triplet and singlet FIG. 2: Spectra for the two Kondo impurity model. The
states of two impurities, so that the results depend on location oftheresonancewith respecttothechemicalpoten-
the spin-correlation S S . For the general case and tial is determined from the fixed point Hamiltonian and the
1 2
h · i width from the leading irrelevant operators. (a) shows the
becauseatleastoneofthetwoverticesinFig.(1),J ,has
+ particle-hole symmetric case; Kondoresonances of both even
singular renormalizations,complete answers can only be
and odd parities appear at the chemical potential. (b) for
obtained by exact methods, for example, Wilson’s NRG
J− =0whilefinitepotentialscatteringV breakstheparticle-
method[9].
hole symmetry in each channel; both resonances appear at
In NRG, the conduction electron band is discretized the same energy but tuned away from the chemical poten-
onalogarithmicgridinto asemi-infinite chainwith only tial. (c) for both finite V and J− while K < Kc; the even
nearest-neighborhoppingterms(herewehavetwochains andoddresonancessplit. For(d),whenK isbiggerthanthe
corresponding to even and odd parity degrees of free- crossover scale Kc, although thesplitting increases, thereso-
dom), while the model can be solved recursively by a nancepeaksfalloutoftheband,reflectingthedisappearance
sequence of H ’s. of the Kondoeffect.
N
H =Λ1/2H + [f† f +H.c.], (6) We find in accordance with the perturbative calcula-
N+1 N X pNσ p(N+1)σ
tion above that the fermi-liquid behavior on either side
p=e,o;σ
3
andthesingularfermi-liquidbehaviorneartheQCPcon-
tinuesforV =0,J =0butwithequaladditionalphase
−
6
shift in both parities so that the asymptotic eigenvalues
are given by 2.0
e,o(a) (a) V=0
V∗f† f , (8) 1.00 (b) V=0.0125
X p 0p 0p e(c) e J-=0
p 12).5 e,o(b) (c) V=0.0125
/ 0.99
with V∗ = V∗. This has a scaling dimension 0 and ( o(c) J-=0.005
e o t -1 0 1 2
appears as a marginally relevant operator at the two 1hif.0
strong-coupling fixed points as well as at the QCP. The es 2.0
s
conservation of axial charge [2, 3] continues to give re- ha 1.5
lation between the coefficients for the irrelevant opera- 0P.5 1.0 o
tors and they diverge in the same manner as at V = 0
0.5
near the QCP. When J = 0, the two impurities be-
− 0.0
comeeffectivelytwonon-interactingKondoscatters. For 0.0 2.35 2.40 2.45
each channel, a potential scattering induced phase shift 0 1 2 0 3 4
δ = tan−1(πρ V)appearsinadditiontotheπ/2phase K/TK
v 0
−
shift, as in single Kondo impurity problem[10]. Here,
δ = δ , two Kondo peaks in local density of states co- FIG. 3: Phase shifts as functions of K. (a)-(c) show the
e o
incide with each other while they are shifted away from cases when V = 0, J− = 0.005; V = 0.0125, J− = 0; and
the chemical potential [see Fig.(2b)]. V =0.0125, J− =0.005, respectively. J+ is chosen as 0.125.
It is natural to represent the eigenvalues in the even
and odd parity channels in terms of the phase-shifts δ
e and δ of two phases are shifted equally from π/2 and
and δ in these channels, which can obtained from the o
o π or 0, respectively. They still behave as two indepen-
eigenvalues of the fixed-point Hamiltonian, denoted as
dentKondoimpurities;eachgainsadditionalδ fromthe
ω∗ for j-th excitation level with p-parity. For free elec- v
pj potential scattering term. With both finite V and J ,
trons(withvanishingV andJ’s),theeigenvalues(η∗)can −
j one can observe two features. 1) the phase shift changes
beeasilyobtainednumericallyforagivenΛ. Theirvalues
smoothlyatacriticalvaluefor K (V), whichsmearsout
at N limit are approximately c
→∞ the critical point. Kc is a function of V since the poten-
1 tialscatteringtermrenormalizestheKondocoupling(see
N odd: η∗ =sgn(j)Λ|j|−1,j = 1,..., (N +1),
j ± ±2 perturbativecalculationabove)thuschangingtheKondo
1 temperature. 2) even and odd phase shifts are different
N even: ηˆj∗ =sgn(j)Λ|j|−1/2,j =0,±1,...,±2N,(9) while their summation remains δe+δo =π+2δv, where
δ can be associated with its counterpart in one Kondo
v
for 1 j N/2. With interactions, the impurityproblem. Inthiscase,althoughtheaxialcharge
eigenvalu≪es(ω|∗)| ca≪n be associated with η∗ by a phase- in each channel is not separately conserved, their sum-
j j
shift factor. Quite generally, mationis. J isarelevantoperatorremovingthecritical
−
point if particle-hole symmetry is absent, V =0.
ωp∗j =Λ−sgn(ηp∗j)δp/πηp∗j. (10) Given the phase shifts the potential scatte6ring param-
eter in the asymptotic Hamiltonian which is a marginal
For instance, at the strong-coupling Kondo fixed point,
operatoraredeterminedthroughδ = tan−1(πρV∗ ),
ωˆ∗ at even iterations (Λj−1/2Λ−1/2) are the same as η∗ e,o − e,o
j j
in odd iterations, corresponding to a phase shift δ =
p (V∗ V∗)(f† f f† f ), (11)
π/2. One can then calculate the phase shifts from the e − o e0 e0− o0 o0
numerical data following Eq.(10). The principal results
which is the coherence operator in effective Hamiltonian
ofthispaperistheevaluationofδ ,δ anditsdependence
e o
onK/TK0 fromwhichthevariationofthecoherencescale Hc∗oh =V12(ψ1†ψ2+H.c.), (12)
is determined. These results are displayed in Fig. (3).
Intheparticle-holesymmetriccase(V =0),δ =δ = where V = D(V∗ V∗)(1+Λ−1)/2, i.e, V in Eq.(3)
e o 12 e − o 12
π/2 in the Kondo-resonance fixed point, but δ =π and for Λ=1.
e
δ =0inthe inter-impurityspin-singletfixedpoint. The The dependence on the coherence scale V on various
o 12
latter merely reflects the fact that in the inter-impurity parameters is shown in Figs.(4,5). V shows a linear re-
12
spin-singlet state, the even parity channel is fully occu- lation with V and J for a whole range of K, as shown
−
piedandtheoddparitychanneliscompletelyempty. An inFig.(5a-b). Thisisinagreementwiththeperturbation
abrupt change of the phase shift happens at the critical results in K limit, while as expected its depen-
point, K 2.4T0, where T0 1.4 10−4D, is the dence on J →for−a∞general K is not. The critical point
c ≈ K K ≈ × +
single-impurity Kondo temperature (for J/D = 0.125). becomesacrossover,asthedegeneracyofthesingletand
J is then irrelevant. When V = 0 but J = 0, δ triplet parts of the Hamiltonian is lifted. Close to the
− − e
6
4
sites. (The analog of this effect in the periodic Ander-
son lattice model, is that the chemical potential falls

within the hybridization gap for the particle-hole sym-
metric case and the system is an insulator [11]). A finite
0.1 V12
o V12 hasimportantimplicationsfortheheavy-fermionlat-
TK
e tice. Heavy-fermionbandshavebeenderivedinavariety
TK
1E-3 of ways, all of which have the same physics: constrained
TK V12
hybridization of a periodic array of local orbitals with a
or wide conduction band with a matrix element of the or-
V121E-5 TK der of the single Kondo impurity resonance width, i.e.,
the Kondo temperature. The heavy-fermion bandwidth
1E-7 insuchcalculationsis thereforeofO(T ). Howevereven
K
for T 0, V gives the order of magnitude of the
K 12
→
splitting of the heavy fermion band at the zone-center
1E-9
-2 -1 0 1 2 3 4 5 6
and the zone-boundary. Therefore, if the lattice prob-
0
K/TK lem is solved by a self-consistent two Kondo impurity
problemwithaclustergeneralizationofdynamicalmean-
FIG. 4: The dependence of V12 and the Kondo temperature field theory, the effective bandwidth is then to be deter-
TK (bothinunitofthebandwidthW)ontheRKKYcoupling mined including both the parameter TK as well as V12.
constantK. OtherparametersarefixedasJ+ =0.125, J− = It is therefore important to consider the relative magni-
0.005 and V =0.0125. tude of these two parameters. The interesting range for
heavy-fermion compounds is for antiferromagnetic cou-
pling K/T0 of O(1). In this range for the typical pa-
K
rameters shown in Fig.(4), V is more than an order
12
1.0 1.0 0.6 of magnitude larger than T . The conclusion then is
K=0 (a) K=0 (b) K=0 (c) that the effect of V dominaKtes in the low energy mass
0.8 J+=0.125 J+=0.125 V=0.0125 12
-3 10 0.6 J-=0.005 V=0.0125 0.4 J-=0.005 aesntdintghetrhmatodtyhnisamteircmanadlrteraadnyspionrctlupdreosperretnieosr.mIatliizsaitnitoenrs-
/ 0. 5 duetospin-correlations,whicharefurtheraugmentedby
V120.4 0.2 Landau-parameters.
0.2 In the regime of large K/TK0, the TK given in Fig.(4)
reflects onlythat the spectralweightatthe chemicalpo-
0.00.00 0.02 0.0400..0000 0.005 0.010 0.00.1 0.2 0.3 tentialisonlyofO(TK/V122). Thisreflectsthedisappear-
V J- J+ anceofthe Kondoresonancefor suchparametersandits
replacement by the spectra schematically illustrated in
FIG.5: ThedependenceofthecoherencescaleV12 onvarious Fig. (2d).
parameters: a) V; b) J−; c) J+ when K =0. Recently, double quantum-dot systems have been fab-
ricated to simulate the two Kondo impurity model [1].
When the two dots are allowed to interact, two split
crossover, although V increases uniformly, the Kondo Kondo resonances appear in the tunneling spectra. The
12
resonances themselves disappear [cf. Fig.(4)]. spectrum in Fig.(3a) of Ref.[1] is similar to our Fig.(2c),
Our most important new physical results come from andthe twoobservedpeakscanbe identified asthe even
the magnitude of V and its dependence on K/T0. In and odd parity resonances of the Kondo dots.
12 K
the particle-hole symmetric case, such a term vanishes: ThisworkwaspartiallysupportedbytheLANL–UCR
the quasi-particle does not hop directly between the two CARE Collaborative Research Programs.
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f
1046(1992). the hybridization vk in the Anderson model, Jk,k′ =
[4] C. Sire, C. M. Varma, and H. R. Krishnamurthy, Phys. 2vkvk′(1/|ǫf|+1/|ǫf+U|)andVk,k′ =(1/2)vkvk′(1/|ǫf|−
Rev.B 48, 13833(1993). 1/|ǫ +U|).
f
[5] B. A. Jones, B. G. Kotliar and A. J. Millis, Phys. Rev. [8] We pick K > 0 to be antiferromagnetic while K < 0 to
5
beferromagnetic, whileinRef.[2]anothernotationI0 is 11, L597 (1978); ibid,12, 3301(1979).
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