physicastatussolidi, 21January2013 Quantum criticality in the two-channel pseudogap Anderson model: A test of the non-crossing approximation FarzanehZamani,*,1,2 TathagataChowdhury,3 PedroRibeiro,1,2 KevinIngersent,3 StefanKirchner1,2 3 1MaxPlanckInstituteforthePhysicsofComplexSystems,No¨thnitzerStr.38,D-01187Dresden,Germany 1 0 2MaxPlanckInstituteforChemicalPhysicsofSolids,No¨thnitzerStr.40,D-01187Dresden,Germany 2 3DepartmentofPhysics,UniversityofFlorida,Gainesville,Florida32611-8440,USA n ReceivedXXXX,revisedXXXX,acceptedXXXX a PublishedonlineXXXX J 8 1 Keywords: Quantumcriticality,Andersonmodel,two-channel,pseudogap,NCA,NRG,scaling. l] ∗Correspondingauthor:[email protected],Phone:+49-351-871-1128 e - r t s t. We investigate the dynamical properties of the two- This model exhibits continuous quantum phase transi- a channel Anderson model using the noncrossing tions between weak- and strong-coupling phases. The m approximation (NCA) supplemented by numerical NCA is shown to reproducethe correct qualitative fea- - renormalization-group calculations. We provide ev- tures of the pseudogapmodel, including the phase dia- d idence supporting the conventional wisdom that the gram,andtoyieldcriticalexponentsinexcellentagree- n o NCA givesreliableresultsforthestandardtwo-channel mentwith the NRG andexactresults. Theformsof the c Andersonmodelof a magneticimpurityin a metal. We dynamicalmagneticsusceptibilityandimpurityGreen’s [ extendtheanalysistothepseudogaptwo-channelmodel functionatthefixedpointsaresuggestiveoffrequency- describing a semi-metallic host with a density of states over-temperaturescaling. 2 thatvanishesinpower-lawfashionattheFermienergy. v 0 5 Copyrightlinewillbeprovidedbythepublisher 4 4 . 1 1 Introduction Quantum criticality is currently pur- ThispaperinvestigatescriticalKondodestructionand 1 sued across many areas of correlated matter, from insu- dynamicalscalinginthepseudogaptwo-channelAnderson 2 lating or weak magnets to metal-insulator systems to un- model. We focus on the treatment of the problem using 1 conventionalsuperconductors.Interestincontinuousphase the noncrossing approximation (NCA), which involves a : v transitionsattemperatureT = 0ismotivatedbothbythe self-consistentevaluationofallirreducibleself-energydi- i appearance of novel phases near such transitions and by agramswithoutvertexcorrections.Withinapseudoparticle X a richness of quantum critical states that extends beyond representationoftheimpurityspin[5],theNCAthreshold r thetraditionaldescriptionintermsoforder-parameterfluc- exponentsoftheT =0pseudoparticlepropagatorsforthe a tuations. A fertile area for experimental study has been one-channelAndersonmodel[6]areknowntodifferfrom the border of magnetism in rare-earth and actinide inter- thecorrectexponentsinferredfromFriedel’ssumrule[7]. metallics, where there is mounting evidence that a cru- Incontrast,theNCAthresholdexponentsformultichannel cial issue is the fate of the Kondo effect on approach to Andersonmodelsagree[8]withthosepredictedbybound- thequantumphasetransition[1,2]:ifKondoscreeningbe- ary conformal field theory [9,10]. The NCA is therefore comescriticalconcomitantlywiththebulkmagnetization, believedtoworkwellforthismodel,whichhasbeenstud- thendynamicalscalingensues,signalingtheabsenceofad- ied extensively as a relatively simple route to non-Fermi ditionalenergyscalesatcriticality[3,4]. liquidbehavior.Thepseudogapversionofthetwo-channel Kondomodel,inwhichthehostdensityofstatesvanishes Copyrightlinewillbeprovidedbythepublisher 2 F.Zamanietal.:Two-channelpseudogapAndersonmodel in power-law fashion at the Fermi energy, was first stud- x-ray edge problem, resulting in maximally particle-hole iedinRef.[11].ThemodelandthecounterpartAnderson asymmetricspectralfunctions: ImG (ω → 0,T = 0) ∼ b modelhaveattractedrenewedinterest[12]duetopropos- Θ(ω)|ω|−αb andImGf(ω → 0,T = 0) ∼ Θ(ω)|ω|−αf. als [13,14] of their realization in the context of adatoms Therefore,G (τ,T)andG (τ,T)shouldberatherdiffer- b f on graphene, where the band structure gives rise to two entfromtheirdynamicallarge-N counterparts,raisingthe symmetry-inequivalentDiracpoints.Recentscanningtun- question of whether dynamical or ω/T scaling—a prop- neling spectroscopydata are in line with expectationsfor erty that arises naturally within the dynamical large-N pseudogapmultichannelKondophysics[15]. approach—cancarryovertotheNCA. Here we briefly revisit the reliability of the NCA for the metallic two-channel Anderson model before turning 3 Asymptotically exact zero-temperature so- tothepseudogapcase,whereweapplyascalingansatzto lution For the case r = 0 of a constant density of obtainT =0thresholdexponents,andhenceextractcriti- conduction-electron states, an exact finite-temperature calexponentsdescribingphysicalproperties.Wefindgood solution can be obtained by transforming the NCA’s in- agreementbetweentheseexponentsandonesobtainedus- tegralequationsintoasetofcoupleddifferentialequations ingfullnumericalsolutionsoftheNCAequationsatT >0 [6]. As this procedure relies on specific properties of the andusingthenumericalrenormalizationgroup. NCA solution for r = 0, its extension to the pseudogap caseisunclear.Wecanaddressthezero-temperatureform 2 The pseudogap M-channel Anderson model of the NCA solution by imposing a scaling ansatz for TheSU(N)×SU(M)Andersonmodelcanbewritten the pseudoparticle spectral functions. The scaling ansatz has been successfully used to extract critical properties N M of impurity models treated within the dynamicallarge-N HMCA = ǫkc†kσµckσµ+ǫf fσ†fσ approximation, including the Kondo model both with a Xk σX=1µX=1 Xσ metallic (r = 0) density of states [17] and with a pseu- +V c† f b† +H.c. , (1) dogap [18], the Bose-Fermi Kondo model [19], and the kX,σ,µ(cid:16) kσµ σ µ (cid:17) pseudogapBose-FermiKondomodel[20].Herewe show that a generalizationof this ansatz to the case of extreme where c† creates a conduction electron of wave vec- particle-hole asymmetry (generated by the exact enforce- kσµ tor k in channel µ and spin projection σ. The repre- ment of the constraint) can be used to extract the critical sentation d† = f†b of the impurity electron creation propertiesofthefixedpointswithintheNCA. σµ σ µ operator in terms of pseudofermion creation combined TheNCAself-energiesforthepseudoparticlepropaga- with slave-boson annihilation faithfully reproduces the torsatrealfrequenciesare[21,22] SU(N)×SU(M) Anderson model provided that the con- straintQ= σfσ†fσ+ µb†µbµ =1isenforcedexactly. Σfreσt(ω)= V2 Z dǫf(ǫ)Acσµ(−ǫ)Grbet(ǫ+ω), (3) WeassumPeaconductPion-electrondensityofstates Xµ ρ(ω)=−π1 ImGc(ω)= 2rD+r+11 |ω|rΘ(D−|ω|) (2) Σbrµet(ω)= V2Xσ Z dǫf(ǫ)Acσµ(ǫ)Grfet(ǫ+ω), (4) thatvanishesattheFermienergyω = 0inamannergov- where the superscript “ret” specifies a retarded function, ernedbyexponentr, takentosatisfy0 < r < 1;thespe- A (ω) = −π−1ImGret (ω) = ρ(ω), and f(ǫ) is the cialcase r = 0 describesa flat(metallic)band.The half- cσµ cσµ Fermi-Diracdistributionfunction.Workingattemperature bandwidthDactsasthebasicenergyscaleintheproblem. T =0,wemaketheansatz[23,7] N = 2 and M = 1, Eq. (1) reduces to the standard Anderson impurity model (with infinite on-site Coulomb A (ω)=a Θ(ω)ω−αp, p=f,b, (5) p p repulsion)whichinturncontainsthespin-isotropicKondo modelasaneffectivelow-energymodel,whileforN →∞ forthepseudoparticlespectralfunctionsA (ω)=−π−1 p various saddle-point approximations can be constructed ImGret(ω), with α beinga thresholdexponentand a a p p p [16,8,17].Thedynamicallarge-NlimitofRef.[17]forthe constant.SubstitutingthesescalingformsintoEqs.(3)and spin-isotropic Kondo model uses a generating functional (4),andapplyingDyson’sequations,oneobtainstheself- equivalentto thatforthe NCA, makingit seem naturalto consistencyconditions expectsimilarresults fromthe two approaches.However, it is important to note that within NCA, the slave-boson αf +αb =1+r, (6) propagator G (ω) is not a Hubbard-Stratonovich decou- sin(πα )sin[π(r−α )]× b f f plingfieldandisthereforedynamicevenatthebarelevel, M sin[π(r−α )] sin(πα ) and that G (ω) couples to the local constraint, which is f + f =0. (7) b (cid:26)N α 1+r−α (cid:27) enforcedexactly.Asaconsequence,G (ω)andthepseud- f f b ofermion propagator G (ω) in the NCA develop thresh- Equation (6) is obtained by matching exponents and Eq. f oldbehaviorreminiscentofthecoreholepropagatorinthe (7) by matching amplitudes. Hereafter, we focus on the Copyrightlinewillbeprovidedbythepublisher pssheaderwillbeprovidedbythepublisher 3 " r < r 0 4 (II) (I) 10 (I) (II) (III) (IV)(V) (III) 2 α (r0,αf0) V 10 f (V) (IV) r > r 1 A(ω,Τ=5 10−8D) 0 f (I) (III) (V) A (ω,Τ=5 10−8D) 10-2 b !! r " V g(ω)=4.5 ω−0.5 10-4 f(ω)=0.45 ω−0.5 Figure 1 Left: Variation with pseudogap exponent r of dpeorsssoibnlemsoodleult.ioDnasshαefdolifneEsqr.e(p7r)esfeonrttuhnesttawbole-cfihxaendneploAinnts- 10-8 10-6 1 0ω-4/D 10-2 1 and dotted lines show values not observed in numerical Figure 2 Pseudoparticle spectral functions A (ω) and f solutionsof thefinite-temperatureNCA equations.Right: A (ω) fora metallic host(r = 0), calculatedat tempera- b Renormalization-group (RG) flows and fixed points as tureT =5×10−8Dforǫ /D=−0.6and(V/D)2 =0.5. f functions of hybridization V for the ranges 0 < r < r0 TheNCApseudoparticlethresholdexponentsα = α = f b andr > r0. Labels(I)–(V)connectRG fixedpointswith 0.5agreewiththeexactlyknownresults. NCAsolutionsintheleftpanel. two-channelcaseN = M = 2,forwhichthepossibleso- 1200 f(T)=-95.0 log(120.0 T) lutionsofEqs.(6)and(7)areplottedschematicallyonthe 1000 r-α planeinFig.1.Forr = 0,thesolutionsareα = 0, f f α = 1 (local moment),and α = 1 (intermediate cou- Τ) 800 plfing),inagreementwithRef.[7f].Fo2r0 < r < r0,where ω=0, 600 thecondition(r0+1)π/2=cot(r0π/2)yieldsanumerical χ( valuer0 ≃ 0.292,there arefive solutions,ofwhichthree 400 correspond to stable renormalization-group fixed points: 200 localmoment(I),two-channelKondo(III),andinfinite-U rtieosnosn,aonftwlehvieclh(oVn)l.yF(oI)rarnd>(Vr)0,artheesrteabalree. just three solu- 0 10-7 10-6 10-5 10-4 10-3 10-2 10-1 T/D The(physical)impurityGreen’sfunctionisobtainedas Figure 3 Static susceptibility χ vs temperature T for a Gret (ω)= dǫe−βǫ Gret(ǫ+ω)A (ǫ) metallichost(r = 0)andǫf/D = −0.6,(V/D)2 = 0.5. dσµ Z fσ bµ TheNCAcapturesthelogarithmicdivergenceofχ(T). (cid:2) −Gadv(ǫ−ω)A (ǫ) , (8) bµ fσ (cid:3) where β = 1/T and “adv” means advanced. The T =0 calpropertiesasitsKondocounterpart,inagreementwith scalingformofthisGreen’sfunctionandofthelocalspin the conclusions of Ref. [12]. This justifies our compar- susceptibility χ can be deduced from the pseudoparticle isonsbelowbetweenNCAresultsfortheAndersonmodel propagatorsby simple arguments,yielding ImGret(ω) ∝ andNRGresultsfortheKondomodel(thesmallerHilbert d |ω|1−αf−αb ≡|ω|−r andImχ(ω)∝sgn(ω)|ω|1−2αf. spaceofwhichallowsgreaternumericalefficiency). TheNRGhasbeenappliedpreviouslytothepseudogap two-channelKondomodel[11,12].Forr < rmax ≃ 0.23 4 Finite-temperature NCA solution At tempera- (somewhatsmaller than the NCA value r0 ≃ 0.292), the turesT >0,theNCAequationsareamenabletoanumer- methodfindsthreestablefixedpointsseparatedbytwoun- icalsolutionontherealfrequencyaxisoverawideparam- stable critical points, precisely analogous to the situation eterrange.Detailsofthenumericalevaluationschemecan shownin Fig. 1. For rmax < r < 1, the NRG yieldsone befound,e.g.,in[24]. criticalpointbetweentwostablefixedpoints,againinone- Resultsforr=0:Asmentionedabove,theNCApre- to-one agreement with the NCA scaling ansatz. Of these dictsthecorrectthresholdexponentsformultichannelAn- fixed points, only those corresponding to (I)–(III) can be dersonmodelswithametallicdensityofstates.Thisisil- accessedforlevelenergiesǫ < 0,theregimeconsidered lustrated in Fig. 2, where the fitted pseudoparticle expo- f in the present paper. (The NCA treatment of cases with nentsagreeverywellwiththeNCAscalingansatzandwith ǫ >0willbereportedelsewhere[20].) the boundary conformal field theory for the two-channel f We have confirmed that the pseudogap two-channel Kondo model [25]. Figure 3 demonstrates that the NCA Anderson model shares the same fixed points and criti- correctly captures the logarithmic divergence in temper- Copyrightlinewillbeprovidedbythepublisher 4 F.Zamanietal.:Two-channelpseudogapAndersonmodel (I) (II) (III) (IV) (V) α 1.0 0.967 0.575 0.183 0.15 10 f α 0.15 0.183 0.575 0.967 1.0 b Table1 PseudoparticlethresholdexponentsatT = 0for thetwo-channelAndersonmodelwithr =0.15. 1 A (ω,T=5 10-8D) d A (-ω,T=5 10-8D) d 108 A(ω,T=5 10-8D) f(ω)=0.31ω−0.166 1046 Ag(fbω(ω)=,T0.=15 ω 1−00-.89D7 ) 10-1 AAddNNRRGG((ω-ω,Τ,Τ==00)) 10 f(ω)=0.83 ω−0.19 -8 -6 -4 -2 1 2 10 10 10 10 10 ω/D 1 Figure 6 ImpurityspectralfunctionA (ω)forr = 0.15 d 10-2 very close to the critical fixed point (II) in Fig. 1, calcu- lated using the NCA at temperature T = 5×10−8D for -4 10 ǫ /D = −0.6 and (V/D)2 = 0.27. Also shown is the f 10-8 10-6 10-4 10-2 1 spectral function calculated within the NRG for T = 0, ω/D ǫ /D =−0.1and(V/D)2 ≃0.0486.TheNCAandNRG f spectralfunctionsaredescribedbyverysimilarexponents Figure 4 Pseudoparticle spectral functions A (ω) and f A (ω) for r = 0.15, T = 5 × 10−8D, ǫ /D = −0.6, (0.166and0.150,respectively),andbothshowthatatthe b f and(V/D)2 =0.27.Thepower-lawbehaviorscorrespond criticalpoint,Ad(ω)isparticle-holesymmetricatlowen- ergies. tosolution(II)inTable1andFig.1. 1 tively vanishes. Obtaining converged numerical solutions nearthisfixedpointisverydifficultastheresultingsharp 10-3 featurescannotberesolvedondiscretefrequencygrids.In- T) T=5 10-2D creasingthehybridizationV untilasolutioncanbestabi- χ(ω, 10-6 T=5 10-3D lized atthe lowestaccessible temperaturesyieldsthe crit- m T=5 10-4D ical solution(II)at V = Vc where (Vc/D)2 ≃ 0.27.Fig. 4 I 10-9 T=5 10-5D 4 shows that the pseudoparticle exponents at this critical 3 0.9T10-12 T=5 10--67D pTohientteamgrpeeerawtuerlelwdeipthenthdeensccealoinfgthaensstaattzicressuuslctespitnibTialibtylere1-. T=5 10 D -8 flectsthefrequencybehavior.Asaresult,thedynamicallo- 10-15 T=5 10 D calspinsusceptibilityχ(ω,T)atthecriticalpointexhibits 10-6 10-3 ω1/T 103 106 109 thedynamicalscalingform(seeFig.5) χ(ω,T)=T−xΦ(ω/T), (9) Figure5 Dynamicalscalingofχ(ω,T)atthecriticalfixed point (II) in Fig. 1, for r = 0.15, ǫf/D = −0.6, and with x = 0.934. It is instructive to compare this result (V/D)2 =0.27. againstothermethods.The NRG is unableto reliably ac- cesstheregime0 < |ω|/T ≪ 1,soitcannotfullytestfor dynamicalscaling. However,we find forthe two-channel ature T of the local spin susceptibility χ. The sublead- Kondo model with r = 0.15 that at the critical point, ing behavior of the impurity spectral function, Ad(ω) ≡ the NRG gives χ(ω = 0,T) ∝ T−x and Imχ(ω,T = −π−1ImGrdet, is also reproduced[8]: Ad(ω)−Ad(0) ∼ 0) ∝ |ω|−y with x = y = 0.930±0.001.Theseproper- |ω|. Taken together, these pieces of evidence indicate tiesareentirelyconsistentwithEq.(9),andshowthatthe pthat the NCA gives qualitatively correct results for the NCA doesanexcellentjobofcalculatingtheexponentx. r=0two-channelAndersonproblem. We notethatthisexponentdeviatessignificantlyfromthe Resultsforr=0.15:Wenowturntonumericalresults leading-ordervaluex=1−2r2 =0.955comingfroman forthetwo-channelpseudogapAndersonmodel,focusing expansionaboutthelocal-momentfixedpoint[12]. firstonthecaser =0.15withǫ =−0.6Dasarepresen- Figure 6 shows the impurity spectral function A (ω) f d tative example of the behaviorin the range of pseudogap very close to the critical fixed point (II) in Table 1. The exponents 0 < r < r0. For this case, Eqs. (6) and (7) power-lawvariationiscompatiblewiththe|ω|−r expected predict five solutions, listed in Table 1. Solution (I) cor- from the scaling ansatz, which coincides with the exact respondsto the local-momentfixed point where V effec- behaviorknowntoholdatallintermediate-couplingfixed Copyrightlinewillbeprovidedbythepublisher pssheaderwillbeprovidedbythepublisher 5 104 1 1 102 Af(ω,T=5 10-8D) ω,A(T)f1100--42 TTT===999 111000---345DDD ω,A(T)b1100--42 TTT===888 111000---345DDD 1 Ag(bω(ω)=,T1.=05 ω 1−00-.588D) 0.58 T10-6 TT==99 1100--67DD 0.57T 10-6 TT==88 1100--67DD 10-2 f(ω)=0.4 ω−0.575 10-8 T=9 10-8D 10-8 T=8 10-8D 10-8 10-6 10-4 10-2 1 10-6 10-3 1 103 106 109 10-6 10-4 10-2 1 102 104 106 108 ω/D ω/T ω/T Figure7 PseudoparticlespectralfunctionsA (ω,T)andA (ω,T)atthefixedpoint(III)inFig.1forr =0.15,ǫ /D= f b f −0.6,and(V/D)2 =2.0.(a)FrequencyvariationatT =5×10−8.(b),(c)Dynamicalscalingatdifferenttemperatures. exponentsα = 0.58andα = 0.575extractedfromFig. f b 10-1 7(a)areinlinewiththepredictionαf =αb =(1+r)/2of T=9 10-2 D the scaling ansatz.Figures7(b)and7(c)demonstratethat ω(,T)d10-2 TT==99 1100--34 DD athree cteommppeartaibtulerewdiethpetnhdeefnrecqesueonfcyAbfe(ωha,vTio)rasnudchAtbh(aωt,dTy)- A 5 T=9 10-5 D namical, or ω/T-scaling ensues. This carries over to the 1 0.T 10-3 T=9 10-6 D impurityspectralfunction,which,asseeninFig.8,iscom- T=9 10-7 D patiblewiththescaling Gret(ω,T)=T−rΨ(ω/T), (10) 10-410-6 10-3 1 103 106 d ω/T This scaling is consistent with the exact result [12] men- tionedabove,i.e.,A (ω,T =0)∝|ω|−r. Figure8 Dynamicalscalingoftheimpurityspectralfunc- d Figure9plotsthestaticlocalsusceptibilityvstemper- tion A (ω,T) at the stable fixed point (III) in Fig. 1 for d ature at the stable fixed point (III). A rather narrow win- r=0.15,ǫ /D =−0.6,and(V/D)2 =2.0. f dow of asymptotic temperaturedependence—presumably a consequence of a strong subleading contribution to 1/χ(ω = 0,T)—gives an exponent x = 0.195 in rea- -0.195 sonable agreement with the scaling ansatz prediction f(T)=4.4 T 102 x=2αf −1=0.15. ) Results for r=0.4: We end this section by consider- T 0, ing a representative case in the range r0 < r < 1. For = ω r = 0.4, the scaling ansatz predicts an asymmetric crit- ( χe ical point described by αf = αb = (1 + r)/2 = 0.7, R 10 the solution corresponding to (III) in Fig. 1. Figures 10 and9showresultsinthevicinityofthiscriticalpoint,ob- tained for ǫ = −0.55D and (V/D)2 = 5.8. The pseu- f doparticleexponentsα = 0.7,α = 0.68extractedfrom 10-8 10-6 10-4 10-2 1 Fig. 10(a) closely follofw the scalbing ansatz, and the dy- T/D namical scaling of A (ω,T) with an exponent0.37 ≃ r d Figure 9 Static susceptibility χ vs temperature T at the [Fig.10(b)]agreeswiththeexactresult[12].Furthermore, stable fixed point (III) in Fig. 1 for r = 0.15, ǫf/D = the exponent y = 0.44 of χ(ω,T = 0) ∝ |ω|−y fitted −0.6,and(V/D)2 =2.0. from the rather narrow frequency window of power-law behaviorinFig.10(b)isinreasonableagreementwiththe value y = x = 2α −1 = 0.4 predicted by the scaling f points [12] and with the NRG results also plotted in Fig. ansatzundertheassumptionofdynamicalscalingandwith 6. Both the NCA and the NRG show the critical spectral x = y = 0.381±0.001givenbytheNRG.Theimpurity functiontobeparticle-holesymmetricatsmall|ω|. spectral function, shown in Fig. 10(c), also appears to be Thestablefixedpoint(III)inTable1canbeaccessed consistentwithdynamicalscaling. by increasing the hybridization beyond the critical value V .Figure7presentsthepseudoparticlespectralfunctions 5 Conclusion We have investigated the reliability c atvarioustemperaturesfor (V/D)2 = 2.0.Thethreshold of the non-crossing approximation (NCA) for the two- Copyrightlinewillbeprovidedbythepublisher 6 F.Zamanietal.:Two-channelpseudogapAndersonmodel 104 AAg(fbω((ωω)=,,TT0=.=3222 11ω00−--707D.D7)) 11002 Ifm(ω ) χ=(0ω.0, 6T ω=8−0 1.404-8 D) T) 10-1 T=2 10-2 D 102 f(ω)=0.27 ω−0.68 1 A(w,d10-2 TT==22 1100--34 DD 1 10-1 0.37T 10-3 TT==22 1100--56 DD -7 10-4 T=2 10 D 10-2 10-6 10-4 10-2 1 10-210-8 10-6 10-4 10-2 1 10-6 10-4 10-2 1 102 104 106 108 ω/D ω/D ω/T Figure10 Resultsforr =0.4inthevicinityoftheasymmetriccriticalpoint(III)inFig.1,calculatedforǫ =−0.55D f and (V/D)2 = 5.8. (a) Pseudoparticle spectral functions A (ω,T) and A (ω,T) at T = 10−8D. (b) Dynamic spin f b susceptibilityχ(ω,T)atT =10−8D.(c)DynamicscalingofthelocaldensityofstatesA (ω,T). d channel pseudogap Anderson model. This was accom- [8]D.L.CoxandA.L.Ruckenstein,Phys.Rev.Lett.71,1613 plished by comparing finite-temperature, finite-frequency (1993). solutions of the NCA equations with asymptotically ex- [9]I.AffleckandA.Ludwig,Phys.Rev.B48,7297(1993). act zero-temperature NCA solutions, with numerical [10]T.A.Costi,P.Schmitteckert,J.Kroha,andP.Wo¨lfle,Phys. renormalization-groupcalculations,andwithexactresults Rev.Lett.73,1275(1994). where available. In contrast to the well-known shortcom- [11]C. Gonzalez-Buxton and K. Ingersent, Phys. Rev. B 57, ings of the NCA for the single-channel Anderson model 14254(1998). with a constant density of states at the Fermi energy, the [12]I. Schneider, L. Fritz, F.B. Anders, A. Benlarga, and M. Vojta,Phys.Rev.B84,125139(2011). NCAcapturessurprisinglywelltheasymptoticlow-energy [13]K. Sengupta and G. Baskaran, Phys. Rev. B 77, 045417 propertiesofthetwo-channelmodel,bothformetallicand (2008). semi-metallic(pseudogapped)hosts.Incasesofapseudo- [14]Z.G.Zhu,K.H.Ding,andJ.Berakdar,Eur.Phys.Lett.90, gap, the results that we have presented for the magnetic 67001(2010). susceptibility and the impurity spectral function are sug- [15]L.S.Mattosetal.(unpublished). gestive of frequency-over-temperature scaling in the dy- [16]N.Readand D.Newns, J.Phys.C:SolidStatePhys.16, namical properties. More complete testing for dynamical L1055(1983). scalingisplannedasfuturework.Finally,wenotethatthe [17]O.ParcolletandA.Georges,Phys.Rev.B58,3794(1998). validation of the NCA treatment of this problem at equi- [18]M.Vojta,Phys.Rev.Lett.87,097202(2001). libriumopensthewayforitsextensiontononequilibrium [19]L.Zhu,S.Kirchner,Q.Si,andA.Georges,Phys.Rev.Lett. steady-stateconditions[20]. 93,267201(2004). [20]F.Zamanietal.(inpreparation). 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