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Kondo temperature for a quantum dot in an Aharanov–Bohm ring C. H. Lewenkopf1 and H. A. Weidenmu¨ller2 1Instituto de F´ısica, UERJ, R. São Francisco Xavier 524, 20550-900 Rio de Janeiro, Brazil 2Max-Planck-Institut fu¨r Kernphysik, D-69029 Heidelberg, Germany 4 We study the Kondo temperature of a quantum dot embedded into one arm of an Aharonov– 0 Bohm interferometer. The topology of a disordered or chaotic Aharanov–Bohm ring leads to a 0 stochastic term in the scaling equation and in the renormalization procedure. As a result, the 2 Kondotemperature displays significant fluctuationsas a function of magnetic flux. n a PACSnumbers: 72.15.Qm,73.63.Kv,73.23.Hk J 6 2 Recently, the transmission phase of electrons passing change little over an energy scale given by the width of throughaquantumdot(QD)embeddedinanAharanov– the Kondo resonance. We also assume that the levels ] Bohm (AB) interferometer has been measured [1]. This ǫ lie so dense (and that their coupling to the leads is l µ l was done in the Kondo regime and over an energy in- so large) that all scattering through the AB ring in the a h terval given by the width of the “odd” valley separating absenceoftheQDissmoothinenergyonthesamescale. - twoconductanceCoulomb–blockadepeaks. TheQDwith The tunneling matrix elements vP between the QD s µ e half–integerspinplaystheroleoftheKondoimpurity[2]. and state µ of the AB ring carry an upper index P = m Theory predicts the phase shift to be π at low tempera- L,R denoting tunneling through the left (right) barrier, . ture [3]. The results are puzzling and do not follow this respectively. The occurrence of two amplitudes for each t a prediction. In the present Letter we show that the T- state µ of the AB ring reflects the topology of the ring. m matrix of the Kondo problem acquires specific prop- We choose the matrix elements vP real and display the TK µ - ertiesduetothetopologyoftheABring. Thesehavenot ABphaseφexplicitly inthe KondoHamiltonian. Gauge d beentakenintoaccountpreviouslyandprofoundlyaffect invariance allows us to put the entire effect of the AB n the dependence of on the AB phase. They serve as phase φ onto the right barrier separating the QD from o K T c oneexamplefortheroleoffinite–sizeeffects[4]andmeso- the AB ring. Whenever the electron leaves (enters) the [ scopic fluctuations [5] which, together with experiments QDthroughthatbarrier,itpicksupthe phasefactoreiφ on semiconductor quantum dots [6] and nanotubes [7] (e−iφ, respectively). We define v =vL+eiφvR. 1 µ µ µ v have rekindled interest in the Kondo effect [8] recently. We focus attention on the Coulomb blockade mid– 3 Implications of our results for the experimental analysis valley region. Then it is justified to use the s–d model 2 are briefly discussed. for the Kondo resonance [8]. (We have convinced our- 5 selvesthatourargumentsholdlikewiseforthe Anderson 1 Model. We consider a QD carrying an odd number of model [8]. However, we have not yet performed a com- 0 electronsintheKondoregime. TheQDisembeddedinto 4 plete analysis of the latter model). In the s–d model, a onearmofanABringthreadedbyaweakmagneticflux 0 spin S~ (which represents the total spin of the electrons Φ. We take account only of the AB phase φ = 2πΦ/Φ . / 0 t Here Φ is the elementary flux quantum. The AB ring ontheQD)iscoupledtothespinsoftheelectronsinthe a 0 AB ring, m is attached to Λ leads labelled j = 1,...,Λ. Without countingspindegreesoffreedomlabelleds= 1/2,lead 2 nd- bjecloawrritehseNFjercmhainsnuerlfsac(et)ralnasbveellresde mao=de1s,w..i.t±,hNejn.erTgihees HK = EC Xνµss′ vµd†µs(~σss′ ·S~)vν∗dνs′ . (1) o c total number of channels is N = jNj. Here EC is the chargingenergy of the QD, and~σ stands v: The leads are separated fromPthe interior of the AB forthe vectorofthe three Paulispinmatrices. The total i ring by fictitious barriers, and the interior from the Hamiltonian is the sum of these terms, X QD by real tunneling barriers caused by the gate po- =H +H +H +H . (2) r tentials. With E the energy, the lead Hamiltonian is H L AB AB−L K a H = Λ H = dE E c† c , with c† Usingstandardtechniques[5,9]weobtainforthenon– L j=1 j j a,s jasE jasE and c the creation and annihilation operators obeying diagonal dimensionless conductance coefficients gjl with {c†jasE,Pcj′a′s′E′} =Pδjj′Pδaa′δsRs′δ(E −E′). The Hamilto- j 6=l of the many–body Hamiltonian H nian H of the AB ring has a discrete spectrum with AB df eigenvalues ǫ with µ = 1,2,... and is given by H g = dE Tr Γ (E)Gr(E)Γ (E)Ga(E) . (3) µ AB jl −dE µs j l = µsǫµ d†µsdµs where {d†µs,dνs′} = δss′δµν. Leads Z (cid:18) (cid:19) h i and AB ring are coupled by the tunneling Hamiltonian Heref istheFermifunction,Gr andGa aretheordinary H P = dE W (E) [c† d +H.c.]. We retarded and advanced equilibrium many–body Green’s AB−L sj µ ja;µ jasE µs assumethattherealtunnelingmatrixelementsW (E) functions of , and [Γ ] =2π W W . P P R ja;µ H j µν a ja;µ ja;ν P 2 Inthe absenceoftheKondoterm,wedealwithapure In the case of Eq. (5), the T–matrix depends non– K single–particle problem, with a Green’s function Gr,a trivially on φ, see Eqs. (4) and (1). TThus, the central 0;µν givenby[Gr0]−µν1 =(E−ǫµ)δµν+(i/2) j[Γj]µν,andnon– question is: How strong is this dependence of TK on φ? diagonalelementsofthespin–independentscatteringma- The theoretical treatment of the Kondo resonance re- trix S(0) (E) = 2iπ W [GPr] W [10, 11]. quires scaling and/or renormalization techniques. Our Hence,jaa;lsb expecte−d, the µcoνndjuac;tµanc0eµνis glbiv;νen by the question can only be answered in this framework. Landauer–Bu¨ttiker formPula. Poor Man’s Scaling. The essential aspects of poor Wereturntothemany–bodycaseandexpandGr (E) man’sscalingemergealreadywhenoneconsidersthema- µν inGr0p(Eow)eTrKsGor0f(EH)Kµνt,owhwerriete Grµν(E) = Gr0;µν(E) + bttroyirxshνevsle′m|aHennKdtGofor0f(ctEuhse)oTHn–Kmt|µhaestiri.inxWtteoremsseuecpdopinardteeso–srstdthaeetreienxsutHemrKnm,agalitfviaoecnn- µ (cid:2) (cid:3) ∞ involving Gr(E). Two processes contribute to this term: 0 TK =HK [Gr0(E)HK]n (4) The scattering of an electron and that of a hole. Aside n=0 from spin–dependent factors, the first process yields X dtheefindeesfitnhiteioTn–mofaStr(i0x),fworetwheritKeognjdloaseffect. UsingTK and vτGr0;τρ(E)vρ∗ = E+|vτ|2ǫ (6) τ τρ τ − X X gjl =Z dE (cid:18)−ddEf (cid:19) Xab (2(cid:12)Sj(a0);lb(E)(cid:12)2+ +Xτρ vτE+1−ǫτh−ijXa;lbWτ;jaFja;lbWlb;ρiE+1−ǫρvρ∗ (cid:12) (cid:12) (cid:12) (cid:12) where[F−1] =δ δ +i W [E+ ǫ ]−1W . (Sj(a0);lb)∗Trs"−2iπ Wja;µ[Gr0 TK Gr0]µνWlb;ν#+c.c. A similar dejcao;lmb posijtlioanb into aµterjam;µinvolv−ingµ onlyµt;hlbe (cid:16) Xµν (cid:17) states in the AB ring and aPremainder applies to hole +Tr 2iπ W (E)[Gr Gr] W 2 . (5) scattering. OurproblemdiffersfromthestandardKondo s − ja;µ 0 TK 0 µν lb;ν ) situation of a magnetic impurity coupled to the conduc- h(cid:12) Xµν (cid:12) i tion band [8] in three main respects, the last two being (cid:12) (cid:12) Three processes contribute to transport through the AB related to the topology of the AB ring: (i) The energies ring,eachrepresentedbyonetermonther.h.s.ofEq.(5): ǫ do not lie in the conduction band but are defined as τ Scattering of an electron through the AB ring without theeigenvaluesofelectronsmovingintheABring. Thus, its passing through the QD, the interference term be- the summation over τ extends to + . (ii) Gr(E) differs ∞ 0 tween the amplitude for that process and the amplitude from the Green’s function for free electrons. This leads for(multiple) visitsofthe QD,andthe squareofthe lat- to the 2nd term on the r.h.s. of Eq. (6). (iii) The v ’s µ teramplitude. Thisstructurediffersfundamentallyfrom involve the AB phase φ. We address these in turn. that for a QD in the Kondo regime connected to two (i)ThecouplingmatrixelementsvL andvR reflectthe τ τ leadswithoutthetopologyoftheABring. Hereonlythe dynamicsofthe couplingofthe QDto the AB ring. The interference term occurs [5]. This is caused by the spe- v ’s are overlap integrals involving the wave function χ τ cial form of the coupling between Kondo resonance and oftheelectronontheQDandthatofstatesτ withmean channels. level spacing ∆ in the AB ring. Two tendencies char- Dependence upon the AB Phase. We compare the de- acterize the behavior of the v ’s. (a) As the number of τ pendence of g on the AB phase φ with that obtained nodes of the state τ in the overlapregion becomes much jl for a simpler problem: An AB ring with a QD that bigger or smaller than that of χ, the overlap is strongly carries a single level at energy E and causes a Breit– reduced. (b) In a diffusive and/or chaotic AB ring, the 0 Wigner resonance [11]. The resulting expression for g states τ within an energy interval given by the Thou- jl (j =l)hastheformofourEq.(5)exceptforthereplace- less energy E , are strongly mixed. We take account of T 6 ment [11] 2iπW (E) [Gr(E) Gr(E)] W both features in terms of a cutoff model. Within an en- − ja;µ 0 TK 0 µν lb;ν → i(γ +η eiφ) (E E +iΓ/2)−1 (γ +η e−iφ). The ergy window of bandwidth D E around the energy ja ja 0 lb lb T − − ≤ amplitudes γ and η are complex and independent of of the state χ, the v ’s are [11] uncorrelated Gaussian ja ja τ energy E and AB phase φ. The only dependence of the distributed random variables with mean value zero and resultingexpressiononthephaseφisthatgivenexplicitly a common constant variance v2 while the v ’s for states τ and that of the totalwidth Γ= γ +eiφη 2. Un- τ outside of this window vanish. For both ballistic and ja| ja ja| der the assumption that the electrons move chaotically diffusive rings E is large compared to the mean level T P and/ordiffusivelyintheABring,thedependenceofΓon spacing∆ of the states inthe AB ring but smallin com- φ is [11] of order 1/√N and, thus, negligible whenever parison to the Fermi energy. The cutoff then applies N 1. The experimental determination of the trans- likewisetothesummationoverparticlesandtothatover ≫ mission phase through the QD (defined as the phase of holes. The cutoff restricts the summation over τ to a the Breit–Wigner term) is, thus, possible for N 1. finite number g E /∆ states, with g 1. To identify T ≫ ≈ ≫ 3 the influence of the largecontributions to Eq.(6) and to the bandwidth and ρ J for the coupling strength such 0 0 resum them, we let g and use scaling. that the relation T = D exp( 1/2ρ J ) holds. We K 0 0 0 →∞ − (ii) Scaling needs to be applied only to the first term thensolveEq.(8)byreducingρ D inunitstepsdrawing 0 ontheright–handsideofEq.(6)sinceonlythistermpro- ψ from a random–number generator. We thereby elim- duces a logarithmic singularity at the upper band edge. inate the states of the AB ring one by one. In the pth As for the second term, the expressionin brackets is ob- step, the coupling constant ρ J is “kicked” out of one 0 viously finite. There are two contributions to the sum- scaling trajectory and into another by the term propor- mations over τ and ρ. For τ = ρ, we sum squares of tional to cosφcosψ . The effect becomes larger as ρ D p 0 amplitudes v but also of the propagator. The result is is decreased and δJ/δD increases. As a result, every re- τ inverselyproportionaltoDandvanishesforD . For alization ψ of the sequence of ψ–values determines a p →∞ { } τ =ρ,weusethatthev ’sareGaussianrandomvariables different scaling trajectory given by a discrete sequence µ 6 with mean value zero. Their random signs quench the ofpoints D ,J . Thescalingprocesshasnecessarilyto p p { } intermediate–state summations and ensure that the re- end when all ring states are reduced to a single remain- sultisconvergentevenasg . Thus,thesingularcon- ingonethatabsorbsallinteractioneffects,givingalower →∞ tributions to the second–order particle matrix element energy cutoff ∆. The lower cutoff is modified for two areobtainedbyreplacingGr(E)by[E+ H ]−1. Cor- reasons: (i) The AB ring is open, and the states µ have 0 − AB respondingstatementsapplytothetermsofhigherorder finite widths with mean value Γ = ∆N/(2π). We have in H . These arguments apply likewise to hole scatter- neglected Γ by omitting the second term on the r.h.s. of K ing. Thus we have reduced our problem to one which is Eq. (6). This is justified only as long as D Γ. (ii) ≫ quite similar in appearance to the standard Kondo case: The electrons which screen the QD spin, form a Kondo Singular terms arise only from the coupling of the QD cloudofsizeξ intheABring. TheenergyscaleE which ξ to the states in the AB ring, i.e., from the Hamiltonian is required to attain a state of correlation length ξ, can H +H . The terms H and H in of Eq. (2) be estimated using the uncertainty relation. Hence, the AB K L AB−L H are irrelevant for scaling. lower cutoff is λ=max ∆,Γ,E . ξ { } (iii) We turn to the AB phase appearing in the We have solved the discretized versionof Eq. (8) for a intermediate–state summation. The first term on the choice of parameters which resembles the weak Kondo right–hand side of Eq. (6) contains the phase through coupling experiment in Ref. [1]: (i) Γ/∆ 10 (Λ = |ivnτg|2va=ria(vbτLle),2w+e(vwτRr)it2e+th2icsoesxφprveτLssviτoRn. Tinotdheefifnoermthe[(vscLa)l2- 6si,zNejL≈of1th0e);riEnξg/).∆(i≈i) E1 (/s∆inc=e√ξ4≈πN1µmwhie≈sreaNbouits tthhee τ T e e +(vτR)2][1+cosφcosψτ] where numberofelectronsonthering. Thisformulafollowsfor ballistic system from E h/L and the Weyl formula. T cosψ = 2vτLvτR . (7) The same formula yields N≈e 1000. (iii) TK/∆ 5. τ (vτL)2+(vτR)2 Our results are shown in F≈igure 1. The inser≈t shows five representative trajectories for cosφ = 1 (where Since the v ’s are uncorrelated Gaussian variables, ψ is τ τ the stochastic effect is largest). The deviations from also a random variable which is uniformly distributed in the standard scaling trajectory increase with decreasing the interval 0 ψ π. Moreover, the ψ ’s with differ- ≤ τ ≤ τ bandwidth, as expected. Averaging over the stochastic ent indices ρ are uncorrelated. We replace the positive trajectories, we recover the standard scaling trajectory quantity [(vL)2+(vR)2] by its average 2v2. An equiva- τ τ (straight line in the insert). In the main part of the lent assumption (independence of the coupling strength figure, we show the distributions P(T ) of Kondo tem- K on the index τ characterizing states in the conduction peratures T for cosφ = 1/2 and cosφ = 1. These are K band) is usually made in the standard Kondo case. We obtained from104 realizationsof ψ . For eachrealiza- checked that this assumption does not affect the generic { p} tion, the final value of T is found by a best–fit proce- K features of our results. dure. Mean value T and dispersion δT are deduced We define the scaling variable J = 4v2/EC and apply fromP(T ). The ahveKriagevalue T is veKrycloseto the the standard algebra of poor man’s scaling [8] to write K h Ki standard scaling value and does not show a significant δJ 2ρ J2 dependence on cosφ. The fluctuations are large, how- 0 = (1+cosφcosψ) . (8) ever,andδT increaseswith cosφ, reachingamaximum δD − D K value of δT / T 0.25 at cosφ=1. K K h i≈ Here ρ = ∆−1 is the density of states µ in the AB The numerical simulations show that the fluctuations 0 | i interferometer. We assume that ρ is constant. Because become larger (smaller) as either ρ T or the lower cut- 0 0 K of the appearance of the random variable ψ, Eq. (8) is off λ are decreased (increased). The reason is that the a stochastic differential equation. To solve it, we first stochastic “kicks” become more effective for smaller val- consider the case without stochastic term and define the uesofT and/orD,seeEq.(8). Ourresultsarelesssen- K “standard scaling trajectory” by choosing a Kondo tem- sitivetoE . AsE isincreased,theslopesofthescaling T T peratureT andinitialvaluesρ D (whereD E )for trajectories become dominated by the upper band edge, K 0 0 0 T ≈ 4 0.8 vac denoting the vacuum, the state 0 is defined as | i | i 0 = ( v 2)−1 v∗d† vac . For n 1, the states | i ν| ν| µ µ µ| i ≥ n would be obtained with the help of Schmidt’s or- 6 | i P P 0.6 thogonalization procedure from the sequence HAB 0 , | i H2 0 , .... (We have shown that it is again justified -1) 4 AB| i )TK J0 ttoerampspirnoxthimearetesutlhtienHgatrmidilitaognoinaanlHHa−mHiltKonbiaynH[8A]Bd)e.pTenhde ( 0.4 r( P 2 upon the matrix elements vµ via expressions of the type v2(ǫ )m,withm=0,1,2,.... Justasforpoorman’s µ µ µ scaling, the averages (fluctuations) of such expressions 0.2 0 10 100 Pare independent of (dependent on) the AB phase. The r D 0 fluctuations are maximal for cosφ = 1 and vanish for cosφ=0. Again,thestochasticpartisexpectedtocause 0.0 a spread in the renormalization group trajectories. The 2 4 6 8 10 12 T trend of the spread with temperature T can be ascer- K tained as follows. The Kondo temperature T defines a K marginalfixedpointoftheproblem,whileT =0isasta- FIG. 1: Distribution of Kondo temperatures TK, for cosφ= 0.5 (narrow distribution) and cosφ = 1 (wide distribution), ble fixed point. All trajectories converge to that point. both for hTKi/∆ = 5. Each histogram contains 104 scaling Therefore, as the sample is cooled down, the stochastic trajectories. Insert: Typicalscalingtrajectoriesforcosφ=1. spreadofthetrajectoriesfirstincreasesmonotonicallyto reach its maximum at T = T . Thereafter, the spread K decreases monotonically and vanishes at T =0. were the stochastic kicks havelittle effect. However,this Discussion. Because of the topology of the AB ring, happensveryslowly: Weobserveonlysmallquantitative the Kondo temperature T of a QD embedded into the K changes as we increase ET by an order of magnitude. ring depends stochastically on the AB phase φ. The re- For the physical interpretation of our result, we say sulting spread in Kondo temperature versus AB phase that an electron is on the QD when it interacts with the is significant, amounting to 20 or even 40 percent. The fixed spin S~ of the Kondo Hamiltonian H . Each term influence of stochasticity is expected to decrease as the K of the perturbation expansion describes a process or a temperatureisloweredbelowT ,andtovanishatT =0. K sequence of processes where an electron leaves the QD, The precise form of the dependence of T on φ can- K moves in the AB ring, and re–enters the QD. Whenever not be predicted for an individual experimental sample. the electron leaves and enters the QD through the same We expect that for T T , this stochastic dependence K ≈ barrier, the result is a factor (vR)2 or (vL)2, as the case seriously limits attempts to determine the transmission may be. Processes of this type are not different from phase through the QD experimentally, although quanti- the ones that are taken into account in the standard ap- tative predictions will have to be based upon a study of proachandleadtothestandardformofthescalingequa- the Anderson model. Such work is now in progress. tion. The factthatthe QDiscoupledtothe AB ringvia This workwaspartlydonewhile oneofus (HAW) vis- two barriers only contributes a factor two and is other- ited Brazil under the auspices of a CAPES – Humboldt wise irrelevant. There are, however,additional processes award. CHL is supported by CNPq and Instituto do where the electron leaves the QD through the left bar- Milênio de Nanociências. HAW acknowledges stimulat- rier andre–entersit throughthe rightone, or vice versa. ing discussions with Y. Gefen, Y. Imry, and Y. Oreg. Such processes correspond to a complete loop through the AB ring, carry the factor vRvL and the AB phase, andareabsentinthestandardapproach. 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