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Nonequilibrium Transport through a Kondo Dot: Decoherence Effects. J. Paaske1, A. Rosch1,2, J. Kroha3, and P. Wölfle1 1Institut fu¨r Theorie der Kondensierten Materie, Universität Karlsruhe, 76128 Karlsruhe, Germany 2Institut fu¨r Theoretische Physik, Universität zu Köln, 50937 Köln, Germany 3Physikalisches Institut, Universität Bonn, 53115 Bonn, Germany (Dated: February 2, 2008) 4 0 We investigate the effects of voltage induced spin-relaxation in a quantum dot in the Kondo 0 regime. Using nonequilibrium perturbation theory, we determine the joint effect of self-energy and 2 vertexcorrectionstotheconductionelectronT-matrixinthelimitoftransportvoltagemuchlarger n thantemperature. Thelogarithmicdivergences,developingnearthedifferentchemicalpotentialsof a theleads,arefoundtobecutoffbyspin-relaxationrates,implyingthatthenonequilibriumKondo- J problem remains at weak coupling as long as voltage is much larger than theKondotemperature. 2 1 PACSnumbers: 73.63.Kv,72.10.Fk,72.15.Qm ] el Electrontransportthroughquantumdotsorpointcon- shiftsthepolewithrespecttotheFermi-energy,replacing - tacts possessing a degenerate ground state (e.g. a spin) 1 by 1 , and in this case E B. The third way to tr is strongly influencedby the Kondoeffect1, providedthe qωuenchω−thBe logarithmistointrIoRdu∼ceafinitedecoherence s dot is in the Coulomb blockade regime. In the linear re- rate Γ, replacing 1 by ω , implying E Γ. at. sponseregime,theKondoresonanceformedatthedotat ThesrelaxationraωteΓ ω=2+ΓΓs2(V,B,T)andItRhe∼asssociated s s m sufficiently low temperature, i.e. at or below the Kondo decoherenceeffects alsoexistinequilibrium. Inthe limit - temperature TK, allows for resonant tunneling, thus re- of vanishing bias voltage and magnetic field, the scale d moving the Coulomb blockade and leading to conduc- Γ tends to a temperature dependent (Korringa) rate3, s n tances near the unitarity limit. This has been observed Γ(0,0,T) T, which vanishes as T 0, allowing for o in various experiments on quantum dot devices2. thse quantu≪m coherent Kondo state to →be formed. In the c [ TheKondoresonanceis quenchedbyeitherlargetem- case of a finite magnetic field and zero temperature, a 1 plaerrgaetubriae,sTvol≫tagTeK,V, largTeKm.aHgnoweteivcefir,eltdh,eBme≫chaTnKismoroaf fBo-rathnedesxpciint-eddepsteantdeeσnt=ra.teI4n, Γdsy,σn(a0m,Bic,q0u)arnetmitaieinssitfipnritoe- v how and why the Ko≫ndo effect is suppressed is qualita- hibitssingularbehaviora↓tω B butitisnotimportant 0 8 tivelydifferentinthethreecases. TheKondoeffectarises for static quantities, where B∼eliminates all relevant sin- 1 from resonant spin-flip scattering at the Fermi energy. gularities. InthecaseofafinitebiasvoltageV,however, 1 Temperature destroysthe resonancemainly by smearing the finite rate Γ(V,0,0) is instrumental to cut off singu- s 0 out the Fermi surface, whereas a magnetic field lifts the laritieseveninstaticquantitiesforT,B 0. TheKondo 4 degeneracyofthelevelsonthedotandtherebyprohibits effectdevelopsonlytoacertainextent,d→ependingonthe 0 resonant scattering. The effect of a bias voltage, V, is ratio V/T . / K t more subtle. It induces a splitting of the Fermi energies Not only for a quantitative description of experiments a m ofthe left, andthe rightlead. However,this splitting af- in the regime V TK, but even for a crude qualitative fects directly only resonant electron scattering from the understanding of≫Kondo physics out of equilibrium, it is d- lefttotherightlead,butnotanyscatteringwhichbegins necessary to identify the correct relaxation rate Γs. The n andendsonthesamelead. Yettheseremainingresonant question,howlogarithmiccontributionsarecutoff,ises- o processesaresuppressedbyadifferenteffect: thevoltage sentialtoderivethecorrectperturbativerenormalization c induces a current which leads to noise and therefore to group description5,6 and to identify regimes where novel v: decoherence of resonant spin-flips. It is the goal of this strong-coupling physics is induced out of equilibrium. i paper to study those decoherence effects in detail. TheimportanceofthebroadeningoftheZeemanlevels X Inperturbationtheory,thesignatureofKondophysics was pointed out three decades ago by Wolf and Losee7 r arelogarithmicdivergencesarisingfrom(principlevalue) a in the contextof the Kondoesquetunneling anomaly ob- integrals of the type served in various tunnel junctions. Incorporating a Kor- D f(ω) D ringa like, T and B dependent, spin-relaxation rate into dω ln (1) Appelbaum’s perturbative formula for the conductance8 ω ∼ E Z−D IR was found to improve the agreement with experiments wheref(ω)isthe Fermifunction, D ahighenergycutoff considerably (cf. e.g. Refs. 9 and 10). Later, in the (i.e. bandwidth)andE someinfraredcutoff. Thereare context of quantum dots, Meir et al.11 pointed out that, IR three rather different ways to cut off the logarithm, and even at T = B = 0, the finite bias-voltage induces a to destroy the Kondo effect, corresponding to the three broadening of the Zeeman levels. In their self-consistent mechanisms discussed above. First, temperature broad- treatmentoftheAndersonmodel,usingthenon-crossing ensf(ω)leadingtoE T. Second,amagneticfieldB approximation (NCA), this nonequilibrium broadening IR ∼ 2 wasshowntosuppresstheKondopeaksinthelocalden- calculation of the cut-off Γ requires a consistent resum- s sity of states, located at the two different Fermi levels. mation of self-energy and vertex corrections. As will be- In Ref. 12 we showed that this NCA relaxation rate is comeclearinthefollowing,thisisaformidabletask,and sufficientlylargetoprohibitthe flowtowardsstrongcou- we have therefore concentrated on the quantity which pling for V T . In a perturbative study of the effects appears to be most tractable: the conduction electron K of an ac-bia≫s, Kaminski et al.13 argued that an irradia- T-matrix as a function of frequency, in zero magnetic tioninducedbroadeningservesto cutoffthe logarithmic field. divergence of the conductance as T and V tend to zero. InSec.Iweintroducethemodelandsomeconventions TreatmentsoftheKondomodel14andrelatedproblems15 used for the Keldysh perturbation theory. A combina- at large voltages, which neglect the influence of deco- tion of self-energy corrections from Sec. IIA and vertex- herence, find strong coupling effects even for V T . corrections calculated in Sec. IIB determines the spin- K Colemanet al.14 recentlyarguedthatthis is the c≫asebe- relaxation rate (Sec. IIC). In Sec. III we show how this cause Γ remains sufficiently small due to a (supposed) decoherence rate cuts off logarithmic corrections in the s cancellation of vertex and self-energy corrections. T-matrix. In Sec. IV we consider the case of anisotropic To our knowledge, even to lowest order in perturba- exchangecouplingsanddeterminetheexactcombination tion theory, a systematic calculation of the nonequilib- oftransverseandlongitudinalspin-relaxationrateswhich rium decoherence rate is still lacking. It is the objective entersthelogarithmsintheT-matrix. AppendicesAand of this paper to provide such a calculation. This is a BcontaindetailspertainingtoSectionsIIBandIII. Ap- delicate matter since self-energy, and vertex corrections pendix C investigates how power-law singularities of the mayindeedcancelpartially,andaninfiniteresummation strongly anisotropic Kondo model are modified out of of perturbation theory is required. Recently16,17, it was equilibrium by mapping it to the nonequilibrium X-ray demonstrated that the Majorana fermion representation edge problem for vanishing spin-flip coupling. forthelocalspin-1/2circumventsthiscomplicationwhen calculatingspin-spincorrelationfunctions. Inthis repre- I. MODEL AND METHOD sentation, such correlatorstake the form of one-particle, ratherthantwo-particle,fermionic correlationfunctions, and consequently only self-energy corrections have to be WemodelthequantumdotbyitslocalspinS~ (S = 1), 2 considered. Whether this representation will prove to coupled by the exchange interaction Jαα′ (α,α′ = L,R) be equally efficient for calculating other observables like to the conduction electrons in the left (L) and right (R) the conductionelectronT-matrix or the conductance re- leads mains to be seen. Based on the conjecture that no unexpected cancel- H = (εk−µα)c†αkσcαkσ−gµBBSz lations occur, we have recently developed a perturba- αX,k,σ tfeivcet raetnolarrmgaelivzoalttiaognegs.rouInpdthesiscraipptpiorona6cohf,tihtewKaosnedsoseenf-- + Jα′αS~ · 21c†α′k′σ′~τσ′σcαkσ, (2) α,α′,k,k′,σ,σ′ tial to include the effects of Γ. For usual quantum X s dots,theKondoeffectissufficientlysuppressedbyΓs6,12, where JLR describes a co-tunneling process transfer- such that renormalized perturbation theory remains ap- ring an electron from the right to the left lead. Here plicable at all temperatures, provided ln(V/TK) ≫ 1. µL,R =±eV/2arethechemicalpotentialsofrespectively We argued that Γs, as a physically observable quantity, the left andrightleads,~τ is the vectorofPaulimatrices, should be identified with the transverse spin relaxation gµ B the Zeeman splitting of the local spin levels in a B rate Γ2 = 1/T2, measuring the coherence property of magnetic field B, and c†αkσ creates an electron in lead the local spin (More precisely, slightly different rates en- α with momentum k and spin σ. We will use dimen- ter into various physical quantities, but to leading order sionless coupling constants gαα′ = N(0)Jαα′, with N(0) in 1/ln[V/TK] one can use Γs Γ2). In this paper we thedensityofstatesperspinfortheconductionelectrons ≈ show that within perturbation theory this is indeed the (assumedflatonthescaleeV,gµ B). Wedefineforlater B case, thus confirming our initial conjecture. Note that use g = (g +g )/2, 4g2 = g2 +g2 +2g2 and d LL RR LL RR LR in more complex situations, for example in the case of use units where ~=k =gµ =e=1. B B coupled quantum dots, Γs can be sufficiently small12 so In order to calculate observable quantities for the sys- that novel (strong coupling) physics can be induced for tem with Hamiltonian (2), we find it convenientto use a large voltages. fermionic representation of the local spin operator, In a preceding paper18, henceforth referred to as I, 1 wtheecdailffceurlaentetdialpceortnudrubcattainvecleyotfhaeKloocnadlomdaogtn,eitnizcalutdioinnganadll S~ = 2 fγ†~τγγ′fγ′, (3) γγ′ leading logarithmic corrections in the presence of finite X V and B. As effects of Γ are not included to this order, with canonical fermion creation and annihilation opera- s some logarithms were not cut off by V but appeared to torsf , f , γ = ,whichallowsa conventionaldiagram- γ† γ ↑↓ diverge with ln(D/T) or ln(D/V B ). A systematic matic perturbation theory in the coupling constant g. | − | 3 Since the physical Hilbert space must have singly occu- II. SPIN LEVEL BROADENING AND SPIN pied states only, it is necessary to project out the empty RELAXATION RATES and doubly occupied local states. This is done by in- troducing a chemical potential λ regulating the charge Thecouplingofthelocalspintotheleadsintroducesa Q= γfγ†fγ. Pickingoutthe contributionproportional broadeningofthe Zeemanlevels,whichdepends ontem- to e βλ and taking the limit λ , the constraint perature, magnetic field and bias voltage. In the pseudo −P → ∞ Q = 1 can be enforced (for a more detailed description fermionrepresentationfor the localspin, the broadening of this method see I). isgivenbytheimaginarypartofthepseudofermionself- We will use the Keldysh Green function method energy. This level broadening enters into the relaxation for nonequilibrium systems, following the notation of rates of both the transverse spin components (S ,S ), x y Ref. 19. Keldysh matrix propagators are defined as whereitaccountsforthelossofphasecoherence,andthe longitudinalspincomponent(S ), whereit describes the GR GK z G= (4) relaxation of the local magnetization following a change 0 GA (cid:18) (cid:19) in the magnetic field. The observable spin relaxation where GR,A and GK are the retarded, advanced and rates,1/T and1/T ,aredefinedthroughthebroadening 2 1 Keldysh component Green functions, respectively. Spec- of the resonance poles in the transverse, and longitudi- tral functions are found as A = i(GR GA), and the nal dynamical spin susceptibilities, and their calculation − greater and lesser functions as requiresvertex correctionsto be included in a consistent way. G>/< =(GK GR GA)/2. (5) ± ∓ Followingabriefdiscussionofthepf self-energybroad- The local conduction electron (ce) Green functions at ening, we determine the renormalized ce-pf interaction thedotintheleftandrightleads,andthepseudofermion vertex in a steady-state nonequilibrium situation. The (pf)GreenfunctionaredenotedbyGab and cd,respec- resultingvertexfunctionsareusedtocalculatethetrans- ασ Gγ tively, with lead index α = L,R, spin indices σ,γ, and versedynamicalspinsusceptibility, andlater,inSec.III, Keldysh indices a,b,c,d. A corresponding notation will they will serve as building blocks for a calculationof the beusedfortheself-energyΣ,anditsimaginarypart,the conduction electron T-matrix. self-energy broadening, is denoted by Γ = i(ΣR ΣA). γ γ − γ The interactionvertexhas the followingtensorstructure in Keldysh space A. Pseudo Fermion Decay Rates 1 Λcd = δ τ1 +τ1 δ , (6) ab 2 ab cd ab cd In paper I (Ref. 18), we determinedthe on-shell imag- where a,b and c,d refer(cid:0)to pf, and ce-li(cid:1)nes, respectively. inary part of the pseudo fermion self-energy, including Since we consider only nonequilibrium situations in a leading logarithmic corrections. For the purpose of this steady state, time translation invariance holds, and the paper, we will only need the second order rates, disre- single-particle Green functions depend only on one fre- garding logarithmic corrections. For T = 0 one finds for quency. The bare pf spectral function is given by 0 V <B: ≤ (ω)=2πδ(ω+γB/2), (7) Aγ Γ = πg2 V, (12) and the Keldysh component Green function is given as ↑ 4 LR K(ω)=i (ω)[2n (ω) 1], (8) Γ = Γ +2πg2B, (13) Gγ Aγ γλ − ↓ ↑ wheren (ω)denotesthepf distributionfunction,given γλ with 4g2 =g2 +g2 +2g2 , whereas for V >B 0: byn (ω)=1/(e(ω+λ)/T+1)inthermalequilibrium. We LL RR LR ≥ γλ shall also use the shorthand notation π Γ = g2 (3V 2B), (14) M =2n (ω) 1. (9) ↑ 4 LR − γλ γλ − Γ = Γ +2πg2B. (15) Assuming a constant conduction electron density of ↓ ↑ states N(0) = 1/2D and a bandwidth 2D, the local ce Noticethatinthepresenceofafinitemagneticfield,only spectral function takes the form the upper spin-level,here correspondingto spin down,is A(ω)=2πN(0)θ(D ω ) (10) broadened when V = 0, as one would expect from sim- −| | ple phase-space considerations. Broadening of the lower with the step function θ(x). The Keldysh component spin-level (spin up) is due to virtual transitions to the Green function in lead α is then given by upper spin-level and occurs only in higher orders in g. GK(ω)= iA(ω)tanh ω−µα , (11) Forcomparison,welistalsothethermaldecayratefor α − 2T V =B =0: (cid:18) (cid:19) assumingtheelectronsineachleadtobeinthermalequi- librium. Γ , =3πTg2. (16) ↑↓ 4 and in general, the convolution of different Keldysh- components gives rise to the polarization tensor = + α′Πd′c′(Ω)= dεGd′c′(ε+Ω)Gdc(ε). (19) α dc 2π α′ α Z It is convenient to form the contraction of this tensor FIG. 1: Vertex equation for the pf −ce vertex, including withtheexchangeconstantsJαα′/4ateachend,andthus define an effective second order interaction by only non-logarithmic irreducible parts. 1 Πd′c′(Ω) J2 α′Πd′c′(Ω). (20) dc ≡ 16 αα′α dc B. Vertex Corrections Contracting again this ce polarization tensor with two bare Keldysh vertices yields the pf interaction tensor Earlywork20,21,22 onthedynamicalmagneticsuscepti- bility of a single spin 1/2,demonstratedhow self-energy, Bba′a′b = Λac′′bdΠdd′cc′Λacbd′′ (21) and vertex corrections combine to yield the transverse, 1 and longitudinal relaxation rates 1/T2 and 1/T1. In = 2 ΠKδa′bδab′ +ΠAδa′bτa1b′ +ΠRτa1′bδab′ , Ref. 20, the vertex corrections were determined in the (cid:8) (cid:9) approximation where the imaginary part of the pf self- where the Langreth rules (cf. Ref. 23) have been em- energy, Γ, is much smaller than temperature. A simi- ployed to work out the contractions lar approach is possible out of equilibrium, where it is Πcd =2ΠK, Πcd =2ΠA, Πcd =2ΠR, (22) the finite voltage,rather than temperature, which deter- dc dc dc mines the abundance of (inter-lead) conduction electron using the notation 1 = 2 and 2 = 1 for the Keldysh particle-hole excitations. In the approximation where indices. As for the single particle Green functions, we Γ V, the dominant corrections to the ce-pf interac- organize these components in a triangular matrix ≪ tion vertex (Keldysh) tensor simplify substantially and the corresponding vertex matrix equation can be solved ΠR ΠK Π= , (23) analytically. Since we assume that V T , perturba- 0 ΠA tion theory is valid and Γ g2V V i≫s indKeed a sound (cid:18) (cid:19) ∼ ≪ and for Ω D one finds that approximation. WeshallconsiderthecasewhereT V, ≪ ≪ whichwillbestrevealthesalientnonequilibriumfeatures π 4Dln2 of the problem. ΠR/A(Ω)=16gα2α′ ±(Ω+µα−µα′)−i π , Tocalculatevertexcorrectionsandtheirinterplaywith (cid:26) (cid:27) stieolfn-edneeprigcyteddiaingrFamig.s,1.wWe hearveemtiondsotlhveertehaedevrerttheaxtepqhuyas-- ΠK(Ω)=π8gα2α′(Ω+µα−µα′)coth Ω+µ2αT−µα′ . (cid:18) (cid:19) ical quantities are proportional to e βλ within our pro- (24) − jection scheme. Therefore we have to keep track of two Noticethatinter-leadparticle-holeexcitationsdonotsat- contributions to the vertex isfy the fluctuation-dissipation theorem as Λãcdb =0Λãcdb+λΛãcdb, (17) α′ΠK(Ω)=coth Ω+µα−µα′ α′ΠR(Ω) α′ΠA(Ω) . α α α 2T − where 0Λ˜ is independent of λ, and λΛ˜ vanishes as e βλ (cid:18) (cid:19)h i − in the limit of λ . We shall first determine 0Λ˜ and The lead-contracted polarization satisfies the following then, in a second→ste∞p, λΛ˜. symmetries: Π>/K/R/A( Ω)=Π</K/A/R(Ω), (25) − and for later use we quote the explicit formula for the 1. Voltage induced Particle-Hole Excitations greater component, Π> =(ΠK +ΠR ΠA)/2: − π Π>(Ω)= g2 [(Ω+V)(1+N(Ω+V)) (26) 8 LR We start by discussing the properties of the ce polariza- +(Ω V(cid:8))(1+N(Ω V))]+2g2Ω(1+N(Ω)) , − − d tion bubble in Keldysh space, which enters as one build- where N(Ω) denotes the Bose-function and Π>(Ω)(cid:9)= 0 ing block in the vertex equation shown in Fig. 1. The for Ω 2D+V. In terms of this function, the second convolution of two conduction electron Green functions ≥ order pf decay rate may be written as has the greater component Γγ(ω)=2θγγ′Π>( ω γ′B/2), (27) α′Π>(Ω)= dεG>(ε+Ω)G<(ε), (18) − − α 2π α′ α with θγγ′ =δγγ′ +2τγ1γ′. Z 5 2. Basic Approximations which is found to have the following components: The following calculations are based on self-consistent γγ′V1cd1 = δcdGγRGγK′ +τc1dGγRGγR′, (30) perturbation theory to order g2 for self-energies and γγ′V1cd2 = δcdGγRGγA′, (31) vertex corrections. As explained in detail in I, self- γγ′V2cd1 = δcd GγKGγK′ +GγAGγR′ consistency is essentialto obtain for example the correct magnetization out of equilibrium. For this paper, we +τc1d(cid:8)GγKGγR′ +GγAGγK′(cid:9), (32) need a self-consistent resummation of diagramsto inves- γγ′V2cd2 = δcdG(cid:8)γKGγA′ +τc1dGγAGγA(cid:9)′. (33) tigate how divergences of bare perturbation theory are cut off to lowest order in the interactions. One proceeds by attaching rungs, using the interaction However, for non-singular quantities like the lowest- tensor Bba′a′b, and legs consisting of pairs of dressed pf propagators. This attachment consists of a contraction order self-energy, self-consistency only gives rise to sub- ofKeldysh,andspinindices,togetherwithanintegration leading correctionswhich we need not keeptrack of. For over the frequency circulating the individual sections of example, it is sufficient to approximate the retarded pf the ladder. To leading order in Γ/V, we may perform propagators (double-dashed lines in the diagrams) by these integrals by neglecting the slow frequency dependence of the ce polarization functions compared to the 1 rapid variations in the pf Green functions. Making use GγR(ω)= ω+γB/2+iΓ /2, (28) of the identity γ 11 1 1 1 = , (34) where Γ , givenin Eqs. (12-15), denotes the on-shell de- ab a b b − a γ − (cid:18) (cid:19) cay rate calculated in bare perturbation theory. We ne- products of Green functions may be expressed as either glectcontributionsfromReΣ (ω)whichcanbeabsorbed γ in a redefinition of B and g, and which give rise only to 1 subleading corrections in the following. GγR(Ω+ω)GγA′(ω)= Ω+(γ−γ′)B/2+i(Γγ +Γγ′)/2 To show, formally, that self-consistency does not 1 1 change this result, one can use the fact that typical in- tegralsaredominatedby integrationsoverfrequenciesin ×(cid:18)ω+γ′B/2−iΓγ′/2 − Ω+ω+γB/2+iΓγ/2(cid:19) a window of width Γ around the Zeeman levels. Since or the various Keldysh components of Π vary slowly with 1 frequency, i.e. [Π(ω+Γ) Π(ω)]/Π(ω) Γ/V, we may R(Ω+ω) R(ω)= therefore use Γ/V ∼ g2 a−s a small expan∼sion parameter Gγ Gγ′ Ω+(γ−γ′)B/2+i(Γγ −Γγ′)/2 (this will be shown explicitly for the vertex corrections 1 1 below). ×(cid:18)ω+γ′B/2+iΓγ′/2 − Ω+ω+γB/2+iΓγ/2(cid:19) and likewise for AR and AA products. Considered as anintegral-kernelto be integratedwith the variouscom- ponents of the polarizationfunction, we may neglect the 3. Summing up the Ladder broadeningandreplaceΩby(γ′ γ)B/2insidetheparen- − theses in such products, and altogether this justifies the approximations ToleadingorderinΓ/V g2,therenormalizedvertex ∼ satisfies the diagrammatic equation depicted in Fig. 1. R(Ω+ω) A(ω) 2πiδ(ω+γ′B/2) , This equation generates a series of ladder diagrams with Gγ Gγ′ ≈ Ω+(γ−γ′)B/2+i(Γγ +Γγ′)/2 dressed pf-legs and bare ce particle-hole propagators as R(Ω+ω) R(ω) 0, (35) rungs. Contributions from diagrams with crossed rungs Gγ Gγ′ ≈ we omit as being of order Γ/V. In appendix A, we es- for a set of legs in the ladder. Notice that Walker20 has tablish this relative smallness explicitly for the crossed employeda similar approximationin the case of thermal 4’th order vertex correction and we expect higher order equilibrium, utilizing the slow frequency dependence of corrections to work in the same way. thethermalce-polarization. Inthiscase,theRRandAA Aniterativesolutionfortherenormalizedvertexstarts terms are neglected to leading order in Γ/T instead. with the attachment of two pf-propagators to the bare Sincethelegscontainnotonlyretardedandadvanced, Keldysh vertex. This defines the tensor but also Keldysh-component Green functions, some of these loop-integrals will also involve the nonequilibrium pf-distribution functions n (ω). This function is found λ γγ′Vacdb =2Λcad′b′Gbγ′aGbγa′′, (29) by solving a quantum Boltzmann equation, obtained as 6 the Keldysh-component of the pf Dyson-equation with each end of the polarization bubble. In zero magnetic second order pf self-energies. Using the results of I, the field this yields the contraction solution at B =0 is found to be τk τi τj τi τj = 2τk , (45) γ′′′γ′′ γ′′γ γ′γ′′′ σσ′ σ′σ − γ′γ n (ω)=n (0)Π<(ω)/Π>(ω), (36) λ λ whichshowsthattheendpointpf Pauli-matrixτk iscar- which, in the case where g = 0 and T = 0, takes the riedthroughtothenewexternalspin-indices. Inthisway, LR 6 form the Pauli-matrix at the endpoint vertex may be left out and the Keldysh vertex merely receives a factor of 2 (gL2L+ggR2L2RR)(ωV+−gωL2R)(V+ω) , 0<ω <V peTrorusnegco.nd order in g, the vertex renormalizes to − n (ω)=n (0) λ λ   gL2R(V−gLω2R)−(V(g+L2ωL)+gR2R)ω , −V <ω <0. 0Λãcdb(Ω+ω′,ω′) (46) aFnodr Tdiv→erg0e,s nasλ(eω)(ωv+aVn)is/hTesfoarsωe−<(ω−VV)./TFoforr ωω ><(3VV7),, = Λacdb−Z d2ωπVacd′b′(Ω+ω,ω)Bba′a′b(ω′−ω) nλ(ω) crudely res−embles a Boltzman−n distribut|io|n with = 21 τc1dδab+δcd τa1b+iτa3a2ΩΠ>+(iωΓ′) , T replaced by V/4. The distribution function clearly (cid:26) (cid:20) (cid:21)(cid:27) inherits the slow frequency dependence from Π> and, wheretheleftsuperscript0istoremindusthatthelimit to leading order in Γ/V, n may therefore be treated λ of λ has been taken. The integral over ω is per- as a constant, when integrated with the rapidly varying → ∞ formed using the δ-function from the RA-product of pf retarded and advanced pf Green functions. In the case Green functions and Γ is the spin-independent (B = 0) of B >0, the distribution function acquires a spin-index single pf self-energy broadening. and the solution is generally more complicated (cf. I). Attaching a set of pf Green functions to this second However, the frequency dependence is still determined order vertex correction, we notice that, after projection by Π>, evaluated at arguments shifted by B/2, and ± and discarding again all RR and AA products, we have therefore remains negligible. In either case, we are thus allowedtoneglectthefrequencydependenceofnλ,which δ τ3 b′a ba′ = Vcd, (47) renders K proportional to R A by a constant and cd a′a′G G − ab G G −G a′b′ reducesallloop-integralsintheladdertoinvolveonlythe X products (35) or their complex conjugates. which in turn implies the fourth order correction Omiting all RR and AA terms, Vcd now simplifies to ab 0Λãcdb(4)(Ω+ω′′,ω′′) γγγγ′′VV11ccdd12 == δδccddMGγRγG′λγAG′,γRGγA′, ((3389)) = 2Z d2ωπ′iΩΠ>+(ωiΓ′)Vac′bd′(Ω+ω′,ω′)Bba′a′b(ω′′−ω′) γγ′V2cd1 = δcd (1−MγλMγ′λ)GγAGγR′ −MγλMγ′λGγRGγA′ = 12δcdτa3a2ΩΠ+>(i0Γ)2ΩΠ>+(ωiΓ′′). (48) +τc1d((cid:8)Mγλ−Mγ′λ)GγAGγR′, (40(cid:9)) From these two lowest order corrections it is clear how γγ′V2cd2 = −δcdMγλGγRGγA′, (41) the further attachment to the ladder will generate a ge- ometric series, and the vertex function andperformingtheprojectionλ ,allpf-distribution →∞ functions vanish, i.e. Mγλ 1, and we are left with 1 1 2iΠ>(ω) →− 0Λ˜cd(Ω+ω,ω)= τ1 δ + δ τ1 +τ3 ab 2 cd ab 2 cd ab aa Ω+iΓ γγ′Vacdb(Ω+ω,ω)=−δcdτb3bGγR(Ω+ω)GγA′(ω). (42) (cid:20) s(4(cid:21)9) therefore solves the diagrammatic equation in Fig.1, in Having performed the projection, it is now a simple the limit λ . We employ the suggestive shorthand mattertosumuptheladdersolvingthevertexequation. →∞ To keep matters simple we assume that B =0, but once 1 Γ = (Γ +Γ )+Γ (50) this specialcaseisworkedout,ageneralizationto B >0 s v 2 ↑ ↓ will be straightforward. We begin by attaching the V- = πg2 V for B,T V, tensor (42) to the ce-polarizationbubble defined in (21). LR ≪ Working out the contraction, one finds that and, as will be demonstrated in the next section, this is indeed the spin-relaxation rate. In the present case Vcad′b′(Ω+ω,ω)Bba′a′b(ω′ ω)= (43) of zero magnetic field and isotropic exchange couplings, − δ τ3 R(Ω+ω) A(ω)Π>(ω ω). (44) the longitudinal, and transverse rates are identical and − cd aaG G ′− thus Γ = Γ = Γ . In the case of anisotropic ex- s 2 1 We should also attach the Pauli-matrices corresponding change couplings (or in the presence of a finite magnetic to the exchange vertices at the endpoint vertex and at field),spin-flip,andnon-spin-flipverticesreceivedifferent 7 corrections and the two rates become discernible. The anisotropic case will be discussed in Sec. IV. The first term in (50) arises from the self-energy, Eqs. (12–15), the second one, Γ = 2Π>(0), is the vertex correction. v Notice that only vertices with identical ingoing and out- going ce Keldysh indices are renormalized. So far, we have only determined the λ limit of the vertex, but we need also the second con→tri∞bution, λΛ˜ inEq.(17),whichisproportionaltoe βλ. Havingsolved FIG.2: Dynamicalsusceptibility. Triangles refertoexternal − for 0Λ˜ already, we are left with the vertex equation measurement vertices. The black (emission) vertex is renormalized like the interaction vertex in Fig. 1, except that the dω twoexternalcelegsareremoved. Theother(absorption)ver- λΛãcdb(Ω+ω′,ω′)= 2 Λ˜ca′d′b′′(Ω+ω,ω) (51) tex remains undressed. − 2π Z b′′a′(Ω+ω) b′a′′(ω)Bba′a′b(ω′ ω), ×G G − C. Dynamical spin susceptibility which we find to be solved by 4Γ λΛãcdb(Ω+ω,ω)=−δcdnλ(0)(cid:26)Ω2+sΓs2Πab(ω) (52) intIrnodourdceerdtionuEnqc.o(v5e0r)t,hweepihnycsluicdaelmheeraenainbgroiefftdhiescruastseioΓns i of the transverse spin susceptibility: + Π>(ω)(τ3 +iτ2 )+Π<(ω)(τ3 iτ2 ) . Ω+iΓ ab ab ab− ab s (cid:27) (cid:2) (cid:3) χR(t)=iθ(t) [S (t),S+(0)] . (57) For c=d one obtains ⊥ h − i 6 i[n (Ω) n (0)] λΛã12b(Ω+ω,ω)= λ − λ Πab(ω), (53) The transverse spin relaxationrate, Γ2, is defined as the Ω iΓs broadening of the resonance pole in this response func- − tion, and as will be shown below, Γ plays exactly this whichis neglecteddue to the slowfrequency dependence s role. Throughout this Section, we may therefore use ofn (Ω). Itisworthnoting,however,thatforB =0this λ 6 Γ = Γ . With a suitable generalization of Γ , entering term will in fact be proportional to the magnetization s 2 v Eq. (50), which will be given in Sec. IV, this identifica- and thus provide an important renormalization of the tion holds also for anisotropic coupling. τ1 term of the interaction tensor. cd Thiscompletesthesolutionofthevertexequationand Translating to the pseudo fermion representation on we maynowproceedto determine its influence onphysi- the Keldysh contour, the transverse susceptibility is cal- calobservables. Indoingso,onehastoattachapairofpf culated from Greenfunctionstotherenormalizedvertex,andmostof- tenonemaythereforecontinuetousetheapproximation (35). Since the dependence of the vertex on the relative ⊥χ(τ)=−i(−i)2hTcK{f↓†(τ)f↑(τ)f↑†(0)f↓(0)}i, (58) frequency ω is set by Πab(ω), one can safely set ω to 0 and consider the vertex as a function of Ω alone. With which in turn leads to the Feynman diagram in Fig. 2 Γ =2Π>(0), the renormalizedvertexthen simplifies to when including vertex, and pf self-energy corrections. v The bare absorption, and emission vertices are given as Λãcdb(Ω)= 12τc1dδab+ 21δcdLab(Ω), (54) aγba1bso=rpt√i1o2nδavberatnedxiγ˜sa1kbe=ptu√1n2dτar1eb,ssreedspaencdtitvheelyem(cifs.siIo)n. Tvehre- tex renormalizes like the interaction vertex-component where L =0L +λL , with ab ab ab √2Λ˜11, whereby ab iΓv Ω+i(Γ+2Γv) 0L (Ω)=τ1 +τ3 iΓv = Ω+iΓs Ω+iΓs dω ab ab aaΩ+iΓs  Ω+iΓ iΓv  ⊥χR(Ω)=i 2πΛã11b(Ω+ω,ω)Gbc(Ω+ω)Gca(ω). (59) Ω+iΓs −Ω+iΓs ab Z ↑ ↓  (55) and Notice that the canonical ensemble average, enforcing single occupancy on the dot, is carried out by dividing 4Γ Πab(0) iΓ λLab(Ω)=−2nλ(0)" Ωs2+Γs2 +τa3bΩ+viΓs#, (56) ttahkeinλg-dtehpeenlidmenittλgrand-ca(ncof.niIc)a.lTahviesrapgreocbeyduhrQeiaλffeacntds → ∞ only the pf distribution functions and allows to neglect wheren (0) e βλ. Usingthisresultwecannowcalcu- all terms proportional to squares, or higher powers of λ − ∝ late physical quantities like susceptibility and T-matrix. n . Working out the contractions, we arrive at γλ 8 dω χR(Ω) = i 0Λ˜11(Ω+ω,ω)2[n (ω) n (Ω+ω)] R(Ω+ω) A(ω) ⊥ 21 2π ↓ − ↑ G↑ G↓ Z n + λΛ˜11(Ω+ω,ω)+λΛ˜11(Ω+ω,ω) 2n (ω)0Λ˜11(Ω+ω,ω) R(Ω+ω) R(ω) 11 21 21 − ↓ G↑ G↓ h i + λΛ˜11(Ω+ω,ω) λΛ˜11(Ω+ω,ω)+2n (Ω+ω)0Λ˜11(Ω+ω,ω) A(Ω+ω) A(ω) . (60) 22 21 21 − ↑ G↑ G↓ h i o The important fact that the final result is proportional similarto aCurie-lawwith 1/T replacedby4/V. Notice to n is ensured by the relations that the result (63), has been obtained also in Ref. 16, γ using a Majorana-fermionrepresentation. 0Λ˜11+0Λ˜11 =0Λ˜11 0Λ˜11 =1, (61) In the case of a finite magnetic field, the factor of 21 11 21 22 − n (ω) n (Ω + ω) in the first term of (60) will be of as such a constant drops after integrating over R R or or↓der B−/V↑. For B max(Ω +B ,Γ), this term will A A. G G therefore dominate t≫he other|terms|involving R R or G G G G In the limit of B 0, the factor of n (ω) n (Ω+ω) A A. For B > 0, the vertex renormalization is modi- in the first term is→of order Ω/V, and↓there−for↑e we are GfiedG, but since we only need to consider the first term in forced to keep also the other terms involving R R or (60), only a single component is needed. For this partic- A A. In this case, we have to keep the full depGenGdence ularcomponentthegeneralizationisstraightforwardand G G of the vertex on two frequencies, but since for example one finds that thepartsofthevertexwhichareretardedwithrespectto ωintegratetozerowith R R,matterssimplifysubstan- 0Λ˜11(Ω+ω,ω)= 1 1 2iΠ>(ω−B/2) , (65) tially. ThefirstsquarebGracGketcansimplybereplacedby 21 2(cid:18) − Ω+B+iΓ2 (cid:19) where Γ is given in Eq. (50) and depends now on both 2 2n (0)Π<(ω) 2Π>(ω)n (ω) V and B (see Eq. (68) below). The integral over ω is λ λ − in (ω) , (62) Ω+iΓ − λ performedusingtheapproximation(35),andthesuscep- (cid:20) 2 (cid:21) tibility is found to be and inserting now the nonequilibrium distribution func- M tion given by (36), the first two terms of this expression χR(Ω) , (66) ⊥ are seen to cancel. We emphasize the fact that this im- ≈ Ω+B+iΓ2 portantcancellationtakesplaceonlywhenusingthecor- valid for max(Ω+B ,Γ) min(B,V) rect distribution function, i.e. the solution to the quan- | | ≪ IntheintermediateregimewhereB min(Ω+B ,Γ), tum Boltzmann equation corresponding to second order ≪ | | one would need to generalize also the λ-dependent part pf self-energies. of the vertex to the case of B > 0. However, we expect The term involving A A works in a similar way, G G that cancellations, similar to those found in terms like and using the approximation R/A(Ω + ω) R/A(ω) (62) at zero field, will take place also at finite B, once G G ≈ ∂ω(ω i0+)−1,valid toleading↑orderinmax↓(Ω,Γ)/V the correct B-dependent distribution function is used. − ± | | when integrated with the slowly varying distribution In this manner, we expect the general formula for the function, the lasttwo terms in (60) may be evaluated by susceptibility to be simply partial integration. The first term comes with a factor aonfd0Λã21l11t(oΩge+thωer,ωo)nGe↑Rfi(nΩd+s tωh)aGt↓A(ω)≈2πiδ(ω)/(Ω+iΓ2), ⊥χR(Ω)≈ MB Ω+BB++iΓ2iΓ , (67) 2 M iΓ valid for max(Ω+B ,Γ) V. This function obviously χR(Ω) 2 , (63) | | ≪ ⊥ has the correct asymptotic behaviors, corresponding to ≈ B Ω+iΓ 2 (63) and (66), and is consistent with the equilibrium for max(Ω,Γ) V. The prefactor is independent of B result20,21. | | ≪ andisobtainedasthederivative n(0),withn(ω)given For completeness, we state here the relevant asymp- ′ − by (37) and with the replacement n (0) 1/2, due to totics of Γ as a function of V, B and T: λ 2 → the normalization by Q before projection. The zero- λ frequency limit obeysh⊥χiR(0) = M/B, like in equilib- πgL2RV , max(T,B)≪V rium, and the nonequilibrium magnetization was found Γ  π(g2 +g2 )B/4 , max(T,V) B in I to be 2 ≈ LL RR ≪ (g2 +g2 +2g2 )B π(g2 +g2 +2g2 )T , max(B,V) T M = LL RR LR , (64) LL RR LR ≪ 2gL2RV  (68) 9 In the equilibrium limit, V = 0, this corresponds to the result obtained in Refs. 20 and 21, Γ 2 ≈ πg2max(T,B/4). + III. CONDUCTION ELECTRON T-MATRIX FIG.3: DiagramsfortheconductionelectronT-matrix,with dressedpf propagatorsanddressedinteractionvertices(black Withtherenormalizedvertexathand,wenowproceed dots). to calculate the conduction electron T-matrix, including the leading logarithmiccorrections. The T-matrix,Tαα′, isofgreatphysicalsignificance,insofarasitdescribesthe scattering of conduction electrons from lead α to lead ′ α, and thereby also the transport across the dot. It is situation, Im[Tαα′σ(ω)] is, at low energies, directly pro- portionalto the spectral function of the electrons on the determinedfromtheconductionelectronGreenfunction: dot (see e.g. Ref. 24 and references therein). This spectral function can be measured directly by tunneling into GRαα′,σ(ω)=GRασ(0)(ω)δαα′+GRασ(0)(ω)TRαα′,σ(ω)GRα′(σ0)(ω). thedot25,andhenceforthweshallfocusontheimaginary (69) part of Tαα′. Incaseswheretheexchange-tunnelingHamiltonian(2)is In Fig. 3 we show the two diagrams contributing to derived from an underlying Anderson model, i.e. from a the T-matrix to third order. Within bare perturbation single quantum dot in the Coulomb-blockade regime (cf. theory (i.e. using bare vertices and Green functions in e.g. Ref.13),onehasJ2 =J J andonlyoneofthe Fig. 3), one obtains the following intra- and inter-lead LR LL RR eigenvalues of the 2 2 matrix Tαα′ is finite. In such a components at T,B =0: × 3π D D Im TR (Ω) = (g2 +g2 ) 1+2g ln +4g g2 ln , (70) αα −16N(0) αα LR αα Ω µ d LR Ω+µ (cid:26) (cid:20) (cid:18)| − α|(cid:19)(cid:21) (cid:18)| α|(cid:19)(cid:27) (cid:2) (cid:3) 3π D D Im TR (Ω) = g 2g 1+2g ln +2(g2 +g2 )ln , (71) LR −16N(0) LR d LL Ω µ RR LR Ω µ (cid:26) (cid:20) (cid:18)| − α|(cid:19)(cid:21) (cid:18)| − R|(cid:19)(cid:27) (cid:2) (cid:3) with µ = µ = V/2. Within bare perturbation the- duce only subleading corrections of order g4, as can be L R − ory,theT-matrixdivergesclosetoeachFermisurface,or shown by an explicit calculation. more precisely, for Ω µα, some of the logarithms are The fate of the logarithms arising to order g3 is more → cut off by the voltageV =µL µR while others remains interesting, and in the following we will therefore care- − unaffected. In this sense voltage and temperature act fully evaluate the second diagram in Fig. 3. This contri- very differently as T would cut off all logarithmic terms bution involves the spin-contractions uniformly. The central question formulated in the intro- ductionis,howthelogarithmicdivergenceswhichremain τk τj τi τi τj τk =24 (72) γγ′ γ′γ′′ γ′′γ σσ′′ σ′′σ′ σ′σ for T 0 and largeV are cut offwhen the perturbation for the Peierls, and → theory is properly resummed. To find the correctcut-off to order g2, we have to replace the bare Green functions τi τj τk τi τj τk = 24 (73) and bare vertices in Fig. 3 by the dressed ones. γγ′′ γ′′γ′ γ′γ σσ′′ σ′′σ′ σ′σ − Asthesecond-orderdiagraminFig.3givesonlyafinite fortheCooper-channel. Writingoutthesumofthesetwo contributionIm[TRαα′(Ω)]=−16N3π(0) α′′gαα′′gα′′α′,the typesofdiagrams,correspondingtodifferentorientations inclusion of self-energy, and vertex corrections will pro- of the pf-loop, one finds that P TRαα(3′)(Ω) = 224(−1)1(i)3(Jαα′′Jα′′α′′′Jα′′′α′/43) d2ωπ 2dπε d2επ′Gαd′′′′c′′(Ω+ε)Gdα′′c′′′(Ω+ε′) Z Z Z × Λ˜1ad′′b′′(ω,ω+ε)Gba(ω)Λ˜ca′b1′(ω+ε′,ω)Gb′a′(ω+ε′)Λãc′′b′d′′′(ω+ε,ω+ε′)Gb′′a′′(ω+ε) nΛ˜c′1(ω,ω ε) ba(ω)Λ˜1d′′(ω ε,ω) b′′a′′(ω ε)Λ˜c′′d′(ω ε,ω ε) b′a′(ω ε) . (74) − a′b − ′ G ab′′ − G − a′′b′ − ′ − G − ′ o 10 This Keldysh contraction has a total of 256 terms, of where A(ε)=1/(ε iΓ)areGreenfunctionsbroadened GΓs − s which only a few will contribute in the end. Some will byΓ ratherthanΓ/2andweusetheshorthandnotation s pinrvoodluvcetaofGm21o,rwehthicahnisonzeerloe,ssaern-dcoomthpeornsenwtill<in,vwolhviecha, Jisα3aαp′′pαa′′′rαe′nt=thJaαtα′t′hJeα′v′αe′r′′tJexα′′c′αo′r.reActliroenasdyhaavtetsheirsvesdtatgoe,reit- G beingproportionaltohigherpowersofthepf distribution place twice the pf self-energy broadening by Γ. Making s function,willvanishfasterthan Q . SincetheKeldysh use of the basic integrals, λ h i representation contains < as part of K = < + >, G G G G it is a daunting task to isolate all contractions with only one factor of <. Nevertheless,since we are dealing here Ddεsgn(ε+a)(ε+b) =ln D2 (81) with a trace oGver the pf Keldysh indices, we are free to Z−D (ε+b)2+Γs2 (cid:18)(b−a)2+Γs2(cid:19) workinamoreconvenientbasisforthe pseudofermions. Thus choosing and 1 1 1 1 ζ = 0 1 , ζ−1 = 0 −1 , (75) D sgn(ε+a)Γ b a (cid:18) (cid:19) (cid:18) (cid:19) dε =2tan−1 − , (82) (ε+b)2+Γ2 Γ theKeldyshmatrixGreenfunctions maybe transformed Z−D s (cid:18) s (cid:19) as representing a broadened logarithm and a broadened Gˆ=ζGζ−1 = G0R 2GA< , (76) εsigann-dfuεncatiroens,trraeisgphetcftoirvwelayr,dt.he remaining integrals over (cid:18) G (cid:19) ′ Thefirstlineoftheintegral(80)involvesaconvolution which has the nice property that ˆ becomes diagonal of GK with A, which yields G after projection. The renormalized vertices may be con- G sidered as functions of only one frequency and therefore take the form (54), which we write loosely as Λ=δ+L. iN(0) ln D2 +2itan−1 Ω−µα′′ . For opposite ce Keldysh-indices the vertex retains the − (cid:26) (cid:18)(Ω−µα′′)2+Γs2(cid:19) (cid:18) D (cid:19)(cid:27) structure of the identity-matrix δ under the transforma- tion. For equal ce-indices, the matrix L transforms to ab This term is multiplied by the convolution of GR with α′ A, equal to iGR(Ω+iΓ), and altogether the first line GΓ α′ Lˆ =ζLζ−1 =0Lˆ+λLˆ, (77) yields the imaginary part where 3π D2 2n J3 N(0)2ln . (83) 0Lˆ = 1 0 , (78) − λ32 αα′′α′′′α′ (cid:18)(Ω−µα′′)2+Γs2(cid:19) φ 1 (cid:18) − (cid:19) and Using a spectral representation for the ce Green functions, the remaining two lines of (80) can be brought to ψ∗ 4ΓsΠR(0) ψ the form λLˆ =2n (0) 2 − Ω2+Γs2 , (79) λ  0 ψ∗ 4ΓsΠA(0)  − 2 − Ω2+Γs2 3 D dε D   2n J3 N(0)2 dω with φ= ΩΩ++iiΓΓs and ψ = Ω2iΓiΓvs. λ32 αα′′α′′′α′ Z−D2π Z−D In this representation th−e contraction in Eq. (74) be- sgn(ε µα′′′) − comes manageable and one has to deal with merely 8 × (ω Ω iΓ)(ω ε iΓ) different types of terms. The full contraction is worked (cid:20) − − s − − s sgn(ε µα′′) out in Appendix B, resulting in + − . (84) (ε Ω iΓ)(ω ε+iΓ) − − s − s (cid:21) 3 dε dε TR(3)(Ω)= n J3 ′ (80) αα′ 16 λ αα′′α′′′α′ 2π 2π Theω integralinfirsttermvanishesinthelimitD , Z Z →∞ × GKα′′(Ω+ε)GRα′′′(Ω+ε′)GΓAs(ε)GΓAs(ε′) asencdonkdeetpeirnmg DbyfianiftaecttohrisotfeΩrm/DreomraΓin/sDsm. aKlleeerptihnagnotnhlye (cid:8)− GRα′′(Ω+ε)GKα′′′(Ω+ε′) the second term, the imaginary parts takes exactly the (cid:2)+GKα′′(Ω+ε)GAα′′′(Ω+ε′) GΓAs(ε)GΓAs(ε−ε′) , same formas (83), and finally we obtainafter projection (cid:3) (cid:9)