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Localization-induced Griffiths phase of disordered Anderson lattices E. Miranda(1) and V. Dobrosavljević(2) (1)Instituto de F´ısica “Gleb Wataghin”, Unicamp, C.P. 6165, Campinas, SP, CEP 13083-970, Brazil. (2)Department of Physics and National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306. We demonstrate that local density of states fluctuations in disordered Anderson lattice models universally leadtotheemergenceofnon-Fermiliquid(NFL)behavior. TheNFLregime appearsat 1 moderate disorder (W =Wc) and is characterized by power-law anomalies, e. g. C/T ∼1/T(1−α), 0 where theexponentαvaries continuously with disorder, as in otherGriffiths phases. ThisGriffiths 0 phase is not associated with the proximity to any magnetic ordering, but reflects the approach to 2 a disorder-driven metal-insulator transition (MIT). Remarkably,theMIT takes place only at much n larger disorder WMIT ≈12Wc, resulting in an extraordinarily robust NFL metallic phase. a PACS Numbers: 71.10.Hf, 71.27.+a, 72.15.Rn,75.20.Hr J 5 The nature of the non-Fermi liquid (NFL) behavior ofstrongelectroncorrelations. (iii)forstrongerdisorder, observed in several heavy fermion compounds remains the NFL metallic behavior persists over a surprisingly ] l largely unresolved [1]. In the cleaner systems, the prox- large interval before a disorder-driven MIT is reached. e imitytoaquantumcriticalpointseemstobeattheorigin This novel Griffiths phase is a manifestation of quan- - r ofmanyoftheobservedproperties[2,3]. Exoticimpurity tum critical behavior associated with the approach to a t s models cannotbe discarded[4],thoughtheir behaviorin disorder-driven metal-insulator transition and does not . t concentrated systems remains ill understood [5]. requirethe proximityofanymagneticallyorderedphase. a m In other compounds, non-stoichiometry has prompted We consider a disordered infinite-U Anderson lattice the investigation of disorder-based mechanisms. A phe- Hamiltonian - d nomenological “Kondodisorder”model (KDM), describ- on ing a broad distribution of Kondo temperatures TK, has H =X(−tij +εiδij)c†iσcjσ +XEfjfj†σfjσ beensuccessfullyappliedtoseveralofthesesystems[6,7]. ijσ jσ c [ Alternatively, the formation of large clusters of magnet- + V (c† f +H.c.), (1) ically ordered material within the disordered phase has Xjσ j jσ jσ 2 alsobeenproposed[8]. Bothscenariosleadtoawidedis- v tribution of energy scales, giving rise to similar thermo- in usual notation. The infinite-U constraint at each f- 9 0 dynamic anomalies and NMR response [7]. In addition, orbitalisassumed(nfj = σfj†σfjσ ≤1). We havestud- 2 thepredictionsoftheKDMprovetobeconsistentwitha ied different types of disoPrder, including randomness in 3 number of other experiments, including optical conduc- theconductionelectronsiteenergiesεi,thef-electronen- 0 tivity [9], magnetoresistance [10] and dynamic neutron ergies Efj, as well as the hybridization Vj. Within our 0 scattering [6,11] measurements. approach, we find that most of our conclusions remain 0 / Despite these successes, a number of basic questions valid for any specific form of disorder, indicating robust t a remain unresolved, including: (1) What is the micro- and universal behavior. m scopicoriginoftheubiquitouspowerlaw(orlogarithmic) We treat the above Hamiltonian within the recently - anomalies? (2) Can a model calculation be done, which proposed statistical dynamical mean field theory [13]. d canproduce these powerlawsin a universalfashion? (3) Thisapproachreducestotheusualdynamicalmeanfield n o Are these properties tied to the proximity to a quantum theory in the limit z → ∞ (with tjk ∼ t/√z) [6,14,15], c phasetransition,andifso,whichone? (4)Howrobustis but unlike the latter, it incorporates Anderson localiza- : the anomalous behavior with respect to the variation of tioneffects. Asaresult,thespectralfunctionofthelocal v i materials parameters? bath “seen” by each impurity has strong spatial fluctua- X Within our model, all these questions find clear-cut tions and contains information coming from sites which r and physically transparent answers: (i) The anomalies are many lattice parameters away. Physically, the fluc- a canbeascribedtoapowerlawdistributionofT ’s,whose tuations of the conduction electron wave-functions lead K exponentvariescontinuouslywithdisorderstrength. The tothedistributionofKondotemperatures,whichinturn resulting NFL behavior, e. g. γ = C/T 1/T(1−α), createsa renormalized effective disorderseenby the con- ∼ α<1setsinforrelativelyweakrandomness,irrespective ductionelectrons. Thisnonlocalfeedbackmechanismre- of the detailed model for disorder. This should be con- sultsinthe universalformofallthe relevantdistribution trastedwiththeKDM[6,7],wheretheoccurenceofNFL functions that we find. behavior requires fine-tuning. (ii) We find universal be- The simplest model for incorporating localization ef- haviorreflectingthenonlocal,many-bodyprocessesasso- fectsisobtainedbyfocusingonaBethelattice ofcoordi- ciated with Anderson localization effects in the presence nation z (with nearest neighbor hopping t, used as unit of energy). The resulting set of self-consistent stochastic 1 equations is governed by the local actions [13,16] β Se(ffj) =Z dτ fj†,σ(τ)(∂τ +Efj)fj,σ(τ)+ 3.0 4 0 Xσ β β dτ dτ′ f† (τ)∆ (τ τ′)f (τ′); (2) (a) 3 (b) Z∆0j(ω)Z=0ω−εXjσ− j,σVzk−=j211t2jjkG(c−jk)(ω).j,σ (3) P(x)12..00 WWW///ttt===001...2555 2P(y)(10)2 WWW///ttt===011...5055 P 1 Here, G(j)(ω) is the local c-electron Green’s function on ck site k with the nearest neighbor site j removed. It is 0.0 0 determined recursively from [13,16] -5 -x4=log-3(T )-2 -1 0 y2=T (410-3)6 8 K K z−1 G(i)(−1)(ω)=ω ε t2 G(j)(ω) Φ (ω), (4) cj − j −X jk ck − j FIG. 1. (a) Distribution of log(TK) for various values of k=1 disorder strength W/t. (b) Distribution of TK showing the Φj(ω)= ω EfjVj2Σfj(ω). (5) efomrmerlgyenwciethofwNidFtLh bWehaavnidorw.eHhearve,eεuis’sedarze=dis3tr,ibEufte=d u−n1i-, − − V =0.5 and µ=−0.5, in units of t. Thelocalself-energyΣ (ω)isobtainedfromthesolution fj oftheeffectiveaction(2)[13,16,17]. Inordertosolvethe Note that the value α = 1, (Figs. 1(b) and 2) impurity problems, we have used the large-N mean-field with P(T ) const., corresponds to the condition for theory at T = 0 [18]. We have solved Eqs. (2-5) nu- K ∼ MarginalFermiLiquidbehaviorobservedinsomeKondo merically by sampling. In implementing this procedure, alloys [6,7], with logarithmically divergentmagnetic sus- we have carried out large-scale simulations for z = 3, ceptibility χ(T) and specific heat coefficient γ. This di- with ensembles containing up to N = 200 sites, and s vergent behavior becomes more singular as the disorder frequency meshes containing up to N = 8,000 fre- omega is increased past this marginal case. For example, if we quencies. Thenumericalintegrationsneededtosolvethe usethesimpleWilsoninterpolationformulaforχ(T)[19] impurity problems have been done by a combination of spline interpolations and adaptive quadrature routines. Λ T(α−1) 1 These careful numerics have made it possible to obtain χ(T)∼Z T +KaT dTK ∼ T(1−α). (7) Kondo temperatures spanning fifteen orders of magni- 0 K tude,whichwascrucialinordertoexaminethelongtails We thus have power law divergences with exponents of the relevant distribution functions. which vary continuously with the disorder strength. If One of the greatest advantages of our approach is its we take t 104K, this should be observed below a few ∼ ability to focus on full distribution functions, which is tens of Kelvin. Such generic behavior has been fitted to essential for characterizing any Griffiths phase. Some some NFL compounds [12]. Besides, χ(0) will diverge at typical results are presented in Fig. 1, where we show a critical disorder strength (W 1 in Fig. 2) c ≈ theevolutionofthedistributionoflocalKondotempera- turesasafunctionofdisorder,fromwhichone computes χ(0) P(TK)dT ΛT(α−2)dT the overall response of the lattice system (See the dis- ∼Z TK K ∼Z0 K K cussion in the first ref. of [6]). We find that (Fig. 1(a)) 1 1 , (8) thedistributionhasauniversallog-normalformforweak ∼ α 1 ∼ W W c − − disorder. Wehaveverifiedthatsuchalog-normalbehav- with a similar result for γ. Note, however, that other ior is obtained irrespective of the type and shape of the higherordercorrelationfunctions,suchasthe non-linear bare disorder distribution, as long as it is not too large. susceptibility χ (0), which probes higher negative mo- As the disorder is increased, the distribution P(T ) 3 K ments of the distribution (χ (0) 1/T3), will begin to nolongerretainsitslog-normalform. Instead,alongtail 3 ∼ K diverge at different critical values of disorder, emergesonthelow-T side,withapowerlawasymptotic K form (Fig. 1(b)) 1 1 χ (0) , (9) 3 ∼ α 3 ∼ W W P(T ) T(α−1). (6) − c3− K ∼ K where W 0.66 for the parameters of Fig. 2. c3 ≈ Theexponentαvariescontinuouslywithdisorder,asseen This general behavior is characteristic of Griffiths on a plot of log(P(log(T ))) in Fig. 2. phases[20]andshouldnotbeconfusedwithatruephase K 2 1.0 2.0 4 3 0.0α 2 WWiitthh of-uet lef-cetlreocntrsons 1.5 1 0 (P(x)) -1.0 0 1W/t2 3 ρtyp1.0 ρtyp=exp(<ln(ρ)>) g π o -2.0 l W/t=0.5 0.5 -3.0 W/t=1.0 W/t=1.5 W/t=2.0 W/t=2.5 -4.0 0.0 -12.5 -7.5 -2.5 0 2 4 6 8 10 12 x=log(T ) W/t K FIG.2. PowerlawasymptoticsofP(log(TK))asthedisor- FIG.3. Localization propertiesoftheconductionelectrons der increases. The linear behavior for small log(TK) implies as monitored by the typical DOS as a function of disorder. a power law dependence of P(TK). Inset: the exponent α of ThesamequantityintheabsenceofKondospinsisshownfor Eq. (6). Same parameters as in Fig. 1. comparison. The vertical dashed line indicates theboundary of the NFL phase. Same parameters as in Fig. 1. transition. The system should be viewed as a disordered metal with embedded clusters of Anderson insula- completely localized. tors. It is precisely these poor conducting regions, with Thisdramaticeffecthasasimplephysicalorigin. Con- depleteddensitiesofstates,whichgiverisetoimperfectly sider the distribution of the effective scattering poten- quenched spins and the corresponding singular thermo- tials of the conduction electrons Φj(ω =0) (see Eq. (5)) dynamic properties. introduced by the f-sites. Note that [6] Φj(ω = 0) = We should also stress that the main mechanism that ZjV2/ε˜fj, where Zj is the quasiparticle weight and − dominates the Griffiths phase is qualitatively different ε˜fj the (renormalized) energy of the Kondo resonance from the one in the KDM. There, T fluctuations were at site j. For sufficient disorder, the Kondo resonances K simplycausedbythedistributionoflocalparameters(V , are randomly shifted up or down in energy, giving rise j Efj)andtheconductionelectronDOSdoesnotfluctuate. to Φj’s that can be random in magnitude but also in Bycontrast,inthepresenttreatment,fluctuationsinthe sign. The resulting distribution for the inverse quantity latter are dominant. To illustrate this, all the results we Φ−j1 is shown in Fig. 4 and is found to broaden with present are obtained for a model with conduction band disorder. For W/t 1.5, a finite density of Φ−1 = 0+ ≈ j disorderonly,althoughsimilarresultsfollowforanyform (i. e. Φ = + ) sites emerges. This is crucial, since j ∞ of disorder. We stress that, in a KDM treatment of this thecorrespondingf-sitesactasunitaryscatterers(US’s), case,T fluctuationsareseverelylimited. Here,however, characterized by a maximally allowed scattering phase K T fluctuations are enhanced by the fluctuations in the shift (δ = π/2) for the conduction electrons. If all the K local conduction DOS, reflecting the localization effects f-sites were US’s, the system would be a Kondo insula- and the approach to a disorder-drivenMIT. tor. The presence of a finite fraction of US’s should be To confirm this picture, we examine the localiza- viewed as the emergence of droplets of a Kondo insula- tion properties of the conduction electrons. We focus tor within the heavy metal. Interestingly, at stronger on the typical DOS ρ = exp < lnρ > ; ρ = disorder (W/t > 4) the distribution of Φ−1(0) contin- typ { j } j j (1/π)ImG (ω = 0), as shown in Fig. 3. This quantity ues to broaden, leading to a decrease in the number of cj vanishes at any disorder-driven MIT [13], and thus can US’s. This is illustrated by plotting P(Φ−1 = 0) in the j serveasanorderparameterforlocalization. Remarkably, inset of Fig 4. In this regime, while the bare disorder we find a strong decrease of this quantity upon entering increases,the effective disorder producedbythe f-sitesis the Griffiths phase (W/t 1), reflecting the strongly reduced, stabilizing the almost localized metallic phase. ≈ enhanced conduction electron scattering due to Kondo This mechanism may be at the origin of the puzzling disorder. Yet, the actual localization transition, where behavior of materials such as SmB [21], where the low 6 the typical DOS vanishes, occurs only at much larger temperatureresistivity remainsanomalouslylargeyetfi- disorder (W/t 12). This results in a very extended nite over a broad range of parameters. ≈ NFL metallic region, where the thermodynamics is sin- Finally, we note that a similar NFL phase was iden- gular, and the conduction electrons are almost, but not tified in a study of the Mott-Anderson transition in the 3 tems and Kondo alloys. 2.0 We acknowledge useful discussions with M. C. Aron- W/t=1.0 2 20 son,D.L.Cox,G.Kotliar,D.E.MacLaughlin,A.J.Mil- W/t=2.0 1 X lis,andG.Zarand. ThisworkwassupportedbyFAPESP 1.5 WW//tt==34..00 =0)1 and CNPq (EM), NSF grant DMR-9974311 and the Al- -1Φj fred P. Sloan Foundation (VD). P( y) 1.0 0 P( 0 2 4 6 8 10 W/t 0.5 [1] Foranoverview,seethepaperspresentedattheConfer- ence on Non-Fermi Liquid Behavior in Metals, J. Phys.: Cond. Matter 8, Number48 (1996). [2] H.v.Löhneysenet al.,Phys.Rev.Lett.72,3262(1994); 0.0 -4.0 -2.0 0.0 2.0 4.0 F.M.Groscheet al.,PhysicaB223+224,50(1996); N. y=Φ-1 D. Mathur et al.,Nature394, 39 (1998) j [3] M.A.Continentinoet al.,Phys.Rev.B39,9734(1989); FIG. 4. Distribution function for the quantity Φ−j1(0) as A. J. Millis, Phys. Rev. B 48, 7183 (1993); A. Rosch et a function of disorder. The inset shows the behavior of al., Phys.Rev.Lett. 79, 159 (1997). the concentration of the strong scattering Kondo centers [4] D. L. Cox, Phys. Rev.Lett. 59, 1240 (1987). (Φ−j1(0) = 0), which reaches a maximum just after the NFL [5] M. Jarrell et al., Phys.Rev.Lett 77, 1612 (1996). phase is entered. Same parameters as in Fig. 1. [6] E.Mirandaetal.,J.Phys.: Cond.Matter8,9871(1996); Phys. Rev.Lett. 78, 290 (1997). [7] O. O. Bernal et al., Phys. Rev. Lett. 75, 2023, (1995); disordered Hubbard model [13]. We have re-examined D. MacLaughlin et al. J. Phys.: Cond. Matter 8, 9855 thissystem,andconcludedthattheNFLbehaviorshould (1996) ; Chia-Ying Liu, et al., cond-mat/9909215. be attributed to a related Griffiths phase rather than a [8] A.H.CastroNetoetal.,Phys.Rev.Lett81,3531(1997). separate thermodynamic phase of the system. Despite [9] L. Degiorgi L et al., J. Phys.: Condens. Matter 8, 9901 the similarities, several features prove dramatically dif- (1996);A.Chattopadhyayetal.,Phys.Rev.B56R2920 ferent. For Hubbard models, the emergence of NFL be- (1997). haviordoesnothaveadramaticeffectonthe conduction [10] A. Chattopadhyay et al.,cond-mat/9805127. [11] M. C. Aronson et al.,Phys. Rev.Lett. 75, 725 (1995). electrons and no US’s emerge. This observation may ex- [12] M. C. de Andrade et al., Phys. Rev. Lett. 81, 5620 plainthestrongcorrelationbetweenthermodynamicand (1998). transport anomalies in Kondo alloys, but not in doped [13] V. Dobrosavljević et al., Phys. Rev. Lett. 78, 3943 semiconductors. 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Our results demonstrate that a well defined Griffiths phase can exist, which is not restricted to the vicinity of any magnetic ordering and yet seems to be consistent with most puzzling features of disordered heavy fermion sys- 4