Strongly correlated multi-impurity models: The crossover from a single-impurity problem to lattice models
aa r X i v : . [ c ond - m a t . s t r- e l ] A ug Strongly correlated multi-impurity models: The crossover from a single-impurityproblem to lattice models
Fabian Eickhoff and Frithjof B. Anders Theoretische Physik 2, Technische Universit¨at Dortmund, 44221 Dortmund, Germany (Dated: August 21, 2020)We present a mapping of various correlated multi-impurity Anderson models to a cluster modelcoupled to a number of effective conduction bands capturing its essential low-energy physics. Themajor ingredient is the complex single-particle self energy matrix of the uncorrelated problem thatencodes the influence to the host conduction band onto the dynamics of a set of correlated orbitalsin a given geometry. While the real part of the self-energy matrix generates an effective hoppingbetween the cluster orbitals, the imaginary part, or hybridization matrix, determines the couplingto the effective conduction electron bands in the mapped model. The rank of the hybridizationmatrix determines the number of independent screening channels of the problem, and allows thereplacement of the phenomenological exhaustion criterion by a rigorous mathematical statement.This rank provides a distinction between multi-impurity models of first kind and of second kind.For the latter, there are insufficient screening channels available, so that a singlet ground state mustbe driven by the inter-cluster spin correlations. This classification provides a fundamental answer tothe question, why ferromagnetic exchange interactions between local moments are irrelevant for thespin compensated ground state in dilute multi-impurity models, whereas the formation of large spinscompetes with the Kondo-scale in dense impurity arrays, without evoking a spin density wave. Thelow-temperature physics of three examples taken from the literature are deduced from the analyticstructure of the mapped model, demonstrating the potential power of this approach. Numericalrenormalization group calculations are presented for up to five site cluster. We investigate theappearance of frustration induced non-Fermi liquid fixed points in the trimer, and demonstrate theexistence of several critical points of Kosterlitz-Thouless type at which ferromagnetic correlationssuppress the screening of an additional effective spin-1 / I. INTRODUCTION
The different competing phases in strongly correlatedelectron systems caused a lot of attention in the last 50years. Heavy Fermions (HF) [1, 2] are a prominent exam-ple for a heavy Fermi-liquid (FL) formation and super-conducting phases [3]. Magnetically ordered phases [4]can either develop out of a heavy FL with very low mag-netic moments, or out of a local moment phase with al-most unscreened magnetic moments [1]. Another promi-nent example are the high-temperature superconductors,where a superconducting dome at finite doping is locatednext to an antiferromagnically ordered Mott-Hubbard in-sulator [5]. External control parameter, such as dop-ing or pressure, have been used to tune between phasesof strongly correlated electron systems at low tempera-tures: strange-metals with non-Fermi liquid (NFL) prop-erties have often been detected [2] in the vicinity of sucha quantum critical point (QCP) [4, 6, 7]. A sufficientunderstanding of such strange-metals, and their originin strongly correlated electronic systems, is still lacking,and the underlying universality of strange-metal behav-ior that develops at a quantum critical phase transitionis still subject of intense theoretical research.The physics of the Heavy Fermions is governed by acompetition between a heavy Fermi-liquid formation dueto the Kondo effect [8], and a magnetic ordering of lo-calized spins due to the RKKY interaction, both medi-ated by the light quasiparticles of the metallic host [4].Since these phases are orthogonal, the Doniach picture [9] suggests that there exists a quantum phase transitionbetween those two phases.This scenario has triggered intensive work on the two-impurity Kondo problem as a simplified model aimingfor a microscopic understanding of a potential quantumphase transition (QPT) between a magnetically orderedand a heavy FL phase. For the two-impurity Kondo prob-lem [10], however, it turned out that ferromagnetic ex-change interactions between the local moments are irrel-evant for the spin compensated ground state, and the an-tiferromagnetic QCP was shown to be unstable [11, 12]:the two singlet fixed points are adiabatically connectedby a continuous change of the conduction electron scat-tering phase.The Doniach picture has already been questioned atthe advent of early approaches to the PAM or the Kondolattice model. Grewe [13] pointed out, that this purelylocal picture neglects the role of the band mediated in-teraction between the local moments. They influence theFermi liquid phase, as well as the formation of mag-netically ordered phases out of a heavy Fermi liquidphase, driven by the residual quasiparticle interactions.The theory of magnetism in such materials must includelocalized and itinerant magnetic order. Furthermore,Nozieres’ exhaustion scenario [14, 15] challenged the no-tion that the heavy Fermi liquid formation can be asso-ciated with individual Kondo effects at each lattice sitein a periodic system, where the Kondo effect is medi-ated by the local density of states of the host conductionelectrons. In spite of the criticism, the Doniach scenarioremains a paradigm [4] even today in illustrations of thepotential origin of complex phase diagrams [16] in HF.While most of the HF exhibit an antiferromagneticallyordered phase, there is a growing number of ferromag-netic HF compounds [17–30]. Recent experiments onsuch ferromagnetic HF revealed strange-metal behaviorwhen the Curie temperature is smoothly suppressed tozero via hydrostatic pressure [28, 29] or chemical pres-sure [30]. Whereas quantum phase transitions in itiner-ant ferromagnets are always of first-order in the frame-work of Hertz-Millis-Moriya theory [6, 7], experiments[28–30] demonstrate the existence of local criticality withKondo-destruction also in ferromagnetic HF. This find-ing proves that the destruction of antiferromagnetism isnot essential for the varied behaviors of strange-metals.New theorys for such ferromagnetic QCPs, aside fromthe first order spin density wave scenario, are, therefore,highly required and may provide new access in the con-text of strange-metals. Impurity cluster of finite size,where the formation of a spin density wave is suppressed,hence, are a good starting point in order to obtain a mi-croscopic understanding why ferromagnetic correlationsin larger correlated cluster can compete with the Kondo-screening, even if such a competition has not been re-ported in two- and three-impurity models [31–34].The quest for a many-impurity problem that is solv-able, and reveals interesting competing phases connectedby a true QCP, triggered the investigation of the frus-trated three impurity spin problems [33–38]. The con-nection to bulk materials, however, remains unclear, al-though it might be very helpful to illustrate the possi-bility of emerging complex phases. A key observation ofthese papers is the central role of magnetic frustrationthat is able to trigger more exotic phases in correlatedmaterials.In this paper, we present an approach that is able toshed some light from a different perspective onto thisold and fundamental question. We start from the con-ventional multi-impurity Anderson model (MIAM) where N f correlated Hubbard atomic sites are hybridizing withWannier orbitals of a single conduction electron band.This includes the well studied single-impurity Andersonand Kondo models [39], and the periodic Anderson model(PAM), where N f is equal to the number of lattice sitesof the host material N L , as two opposite limits, as wellas finite size impurity clusters that become relevant forscanning tunneling microscopy (STM) or as toy modelsfor magnetic frustration. We present a mapping for theoriginal model to an effective low-energy MIAM that isjustified in the wide-band limit. The mapping accountsfor the conduction band mediated RKKY interaction andthe delocalization of the correlated orbitals by effectivehopping matrix elements between all orbitals, as well asthe Kondo effect by the construction of effective bandchannels.The number of effective screening conduction bandchannels in the mapped model depends on the latticegeometry and the location of the impurities. The num- ber of k-points on the Fermi surface of the host mate-rial provides the upper limit of the screening channels,challenging the Doniach scenario of a Kondo screening ofindividual local spins by a single conduction band in thePAM. As a consequence of the mapping, the magnetic or-dering, the heavy FL formation as well as the screening ofthe local moments are related to collective phenomena,involving a small number of Kondo screening channelsand the conduction band mediated effective interactionbetween the impurity orbitals.In Refs. [29, 40] the paramagnetic-ferromagnetic tran-sition in the Kondo lattice was studied within a inde-pendent bath approximation for each spin, such that thesuppression of the Kondo temperature can be ascribedto the Kondo resonance narrowing in FM coupled single-impurity Kondo models [41] in combination with an in-finite number of coupled local moments. However, dueto the independent bath approximation, the exhaustioneffect of the conduction electrons is neglected in suchmodels.Within the classification we present below, we proposea different mechanism leading to FM criticality in multi-impurity models, directly based on the reduced numberof conduction band screening channels. From the two-and three-impurity problem [31–34] it is known, that FMcorrelations do not compete with the single-ion Kondo ef-fect and lead to a reduction of the Kondo temperature atmost. However, if the number of impurities is large, suchthat there are not enough conduction screening channelsavailable, the singlet ground state can not be interpretedin terms of the single-ion Kondo effect any longer, andadditional collective mechanisms need to be taken intoaccount. We demonstrate that it is this collective sin-glet formation that competes with the formation of FMcorrelations between the local moments.Since the delocalization and the collective screening ofthe individual local moments is both realized by oper-ators responsible for the antiferromagnetic part of theRKKY interaction, ferromagnetic couplings lead to acompetition between (localized) magnetic order and the(delocalized) heavy FL. Consequently, the competitionis rather between (delocalizing) AF and (localizing) FMRKKY interaction, than between (delocalizing) Kondoand (localizing) RKKY coupling, as usually assumed[29, 40].We demonstrate the formation of a ground state withfinite magnetic moment and ferromagnetic spin-spin cor-relations between the local moments in multi-impuritymodels belonging to the class that adiabatically evolvesto the PAM for a large number of correlated orbitals.Due to the finite number of correlated orbitals a spindensity wave scenario can be excluded in this case. Thisferromagnetic type of ground state is beyond the scopeof the generic two impurity model where a spin singlet isalways formed at sufficient low temperatures. The tran-sition between a spin-singlet and a spin-full ground stateis accompanied by a QCP, at which ferromagnetic cor-relations lead to a suppression of the screening of an ef-fective spin-1 / C symmetric setup, and review the frustration inducedNFL fixed points [33, 34] within our effective low-energymodel. While at intermediate strengths of AF RKKYinteraction K RKKY > K c, the NFL fixed point isstabilized, we establish the existence of an upper bound K c, , at which the NFL fixed point gets unstable, andthe system becomes a FL at low temperature. Since theRKKY interaction needs to dominate over the Kondotemperature T K in order to allow for magnetic frustra-tion, the NFL fixed point completely disappears in thephase diagram, if the Kondo temperature exceeds thisupper bound T K > K c, .One of the strengths of our effective low-energy map-ping is that it incorporates the FM and AF RKKY inter-action as well as the generated potential scattering terms:It naturally incorporates the correct symmetries of theoriginal multi-impurity models coupled to only one singleconduction band [11, 12, 32]. We do not need to add arti-ficial Heisenberg exchange couplings, which might lead tounphysical fixed points as known from the two impuritymodel [11, 12, 32], to realize and explore the competingphases.The paper is organized as follows. After introducingthe precise definition of our model in Sec. II A, we pro-vide a preliminary overview of the results in Sec. II B.While the mapped low-energy MIAM is derived in Sec.II C, we introduce the rank of the interaction matrix asquantitative classification in multi-impurity problems offirst and second kind in Sec. II D and discuss severalimpurity-cluster configurations in different spatial dimen-sions. The strength of our mapping is demonstrated inSec. II F where we revisit three different problems inves-tigated in the literature using sophisticated methods andpredicted the central result of each problem: (i) The fer-romagnetic ground state of the dilute PAM [42, 43] athalf-filling, (ii) ferromagnetic ground state in the one-electron limit of the Kondo-lattice model [44], as well as(iii) the scaling of the critical U c of the Mott transitionin the PAM with nearest-neighbor hybridization found inan elaborate dynamical mean field calculation [45]. OurNRG results on three, four and five impurity clusters arepresented in Sec. III. In Sec. III A we study the impuritytrimer in a C symmetric setup, and review the frus-tration induced NFL fixed points [33, 34]. For short 1dimpurity chins in Sec. III B, we report on a sequence ofKosterlitz–Thouless-type phase transitions as function ofthe host band filling and the strength of Coulomb interac-tion. In Sec. III B 1 we study dense impurity arrays, suchthat the fixed point evolves from a singlet at half-fillingto a maximally polarized multiplet at the band edge, and we discuss and explore the role of magnetic frustrationat intermediate band fillings in Sec. III B 2. For dilutemulti-impurity models in Sec. III B 3 the situation is viceverse, i.e. starting from maximally polarized multiplet athalf-filling we can drive the system across several QCPsto a spin-singlet ground state. We conclude the paperwith a short summary and an outlook in Sec. IV. II. THEORYA. Model
Although strongly correlated electron systems have alarge number of incarnations in particular when appliedto realistic material science, we focus on the most elemen-tary version in this paper that targets impurity clusterson surfaces as well as the elementary modeling of HF.These models can be easily generalized to more complexsituations if need, for instance to multiple correlated 3 d -orbitals as required in transition metal ions, but showalready rich physics that is worth presenting from a dif-ferent perspective.Quantum impurity systems are typically embedded ina metallic host which is represented by an non-interactingtight-binding model H host = − X i,jσ t ij c † i,σ c j,σ = X ~kσ ǫ ~kσ c † ~kσ c ~kσ (1)that is diagonalized in k -space in a periodic lattice. i, j label all lattice points ~R i ∈ T L where T L defines the setof all lattice points. The dimension of T L is N L . ǫ ~kσ de-notes the band dispersion obtained from Fourier transfor-mation of the matrix t ij , and can also include a Zeemanterm due to an external magnetic field not considered inthis paper. ǫ ~kσ becomes a continuous function of ~k for N L → ∞ . The diagonal element t ii accounts for the lo-cal orbital energy and is used to shift the band center ofthe conduction band. In this paper, we restrict ourselvesto nearest neighbor tight-binding models for keeping theparameter space simple, but our approach is applicableto arbitrary dispersions ǫ ~kσ .The N f impurities are located at the positions ~R l ∈ T f and are modeled by an atomic Hubbard Hamiltonian H corr = X l,σ ǫ flσ f † l,σ f l,σ + 12 X l,σ U l f † l,σ f l,σ f † l, ¯ σ f l, ¯ σ , (2)where f ( † ) l destroys (creates) an electron in the single-impurity orbital at site l . The on-site energies are labeledby ǫ fl , ¯ σ = − σ , and U l denotes the on-site Coulomb repul-sion. In general, correlated 3 d or 4 f -shells contain manymore degrees of freedom. Here, we focus on the essentialsto kept the number of free parameter to a minimum. Wehave situations in mind where crystal electric fields sepa-rates the ground state doublet energetically from higherexcitations and spin-orbit coupling between the conduc-tion electrons and the local degrees of freedom can beneglected. However, the mapping and the classificationintroduced below is applicable to an arbitrary numberof orbital degrees of freedom as well. The mapping in-troduced below is applicable to arbitrary locations, butthroughout the paper we focus on a finite dimensionalsubset T f ⊂ T L of the underlying lattice.Since we are only considering the thermodynamic equi-librium, we can either explicitly use a chemical potential µ to adjust the different fillings, or we absorb µ by collec-tively shifting all single particle energies, t ii and ǫ fl , bythe same amount and leave µ = 0. We adapt the laterconvention and investigate the effect of different conduc-tion band fillings by shifting the band center ǫ c = t ii .The most general coupling between the two orthogonalsubsystems is given by the spin-diagonal hybridizationterm H hyb = X l,mσ V m,l c † mσ f lσ + h . c . (3)= X ~k,lσ V ~k,l c † ~kσ f lσ + h . c . where V ~k,l is obtained by a Fourier transformation, V ~k,l = 1 √ N L X m V m,l e − i~k ~R m . (4)In this paper, we only consider a local hybridization,i. e. V m,l = δ ml V l , V ~k,l = V l exp( − i~k ~R l ) / √ N L , and anearest neighbor hybridization V m,l = V for ~R m and ~R l being nearest neighbor sites and V m,l = 0 otherwise,corresponding to V ~k,l = − ( V /t ) ǫ ~k [45]. The strengthof the coupling is typically discussed in terms of Γ ,l = πV l ρ (0), which describes the effective hybridization of asingle-impurity with a conduction band density of states(DOS) ρ ( ǫ ).The total Hamiltonian of the system is given by H = H host + H corr + H hyb . (5)This formulation includes two well established and wellunderstood limits. If T f = T L , and N f = N L → ∞ , werecover the PAM. If T f only contains a single site, themodel is known as single-impurity Anderson model thatwas accurately solved using the NRG [46, 47] and theBethe ansatz [48, 49] almost 40 years ago. If the numberof sites 1 < N f ≪ N L is small and finite, we refer to amulti-impurity Anderson model (MIAM) whose simplestrealization is the two-impurity Anderson model (TIAM)[31, 32]. B. Preliminaries
At the heart of this paper lies the extension of the low-energy mapping developed for the two-impurity model
FIG. 1. Schematic partitioning of the models in four differentcategories: I denotes the limit of a single-impurity problem,section II the MIAMs of the first kind, section III the MIAMsof the second kind and section IV the MIAM with infinitenumber of correlated lattice sides recovering the periodic An-derson model. [32] to the multi-impurity situation ( N f >
2) and theconsequences that can be concluded from this mapping.The single-impurity problem [50] as well as the two-impurity problem [11, 12, 31, 51] has been extensivelyinvestigated over the last four decades and are well un-derstood.The interest for the two-impurity problem originatesin the Doniach scenario [9] for Heavy Fermions (HF) [1]which relates the origin of the magnetic ordering found insome of the HFs to the competition between the single-ion Kondo effect [8] screening the local moments andthe RKKY interaction [52–61] favoring magnetic orderingof those moments. Although lacking a rigorous mathe-matical prove, such appealing visualizations of the com-plex physics in HF [4, 16, 62] are popular even today[62], since it provides a simple picture that can intu-itively be grasped. This picture was already challengedby Nozieres’ exhaustion scenario [14, 15], as well as by theobservation that HF magnetism can form even out of aheavy FL phase but with strongly reduced magnetic mo-ments [1, 13]. This indicates that the Doniach scenariois too simplistic and does not reflect the full physics insuch complex correlated electron systems.Figure 1 summarizes the four categories of the MIAMwhich we can mathematically rigorous distinguish withinour mapping presented below. The two well establishedlimits of the MIAM model, the single-impurity Andersonmodel ( N f = 1) and the periodic Anderson model ( N f = N L ) are located at the opposite end of the figure, wherethe horizontal axis denotes the number of impurities N f .We define the MIAM of the first kind by our abilityto map the low-energy physics problem onto an effectivecoupled multi-impurity cluster that couples the N f lo-calized orbitals to N b = N f effective conduction bands:The number of effective conduction bands, N b , exactlymatches the number of impurities which allows for spin-singlet formation by a compensated multi-channel Kondoeffect [63].The first example of such a MIAM of the first kindis the well understood two impurity model [10–12, 31,51, 64], N f = 2. Jones and Varma showed [31] thatthe model can be mapped onto a two-impurity, two-bandmodel in the even/odd parity basis. The induced or-bital hopping [32] is responsible for the antiferromag-netic (AF) exchange, the asymmetry of the couplings tothe two bands results in the ferromagnetic (FM) part ofthe RKKY interaction [31], while the two bands allowto screen the impurity moments via a two-stage Kondoeffect. The QCP that emerges, if the energy dependenceof the hybridization functions in the even/odd basis isneglected [10], is just a consequence of unphysical ap-proximations [11, 12] which automatically restores a spe-cial kind of particle-hole symmetry that is absent in theoriginal model. Since the antiferromagnetic RKKY in-teraction, which is required to drive the phase transi-tion, in the full model is dynamically generated from thesame contributions that break this special symmetry, theQCP is replaced by a continuous crossover once the fullenergy dependence is correctly incorporated. Any ap-proximative solution of multi-impurity problems, as toymodel with regard to quantum criticality in HF there-fore, need to ensure the absence of a QCP in the twoimpurity limit. Other examples of MIAM of the firstkind are trimer models [33–35] involving three effectiveconduction bands.Depending on the details of the lattice topology andthe geometric arrangement of the impurities, we find acritical value N cf ( T L , T f ) above which the MIAM mapsonto a low-energy multi-impurity cluster that couples toa reduced number of effective conduction bands N b < N f .We call those type of problems MIAM of the second kind,indicated by the category III in the Fig. 1. This reducednumber of coupled conduction bands has profound conse-quences for the magnetic properties of the system: a largelocal moment, that is formed at low temperatures, can-not be completely screened by the multi-channel Kondoeffect. We argue below that the periodic Anderson model(PAM) is an particular example for such an MIAM prob-lem of the second kind: The screening of the local mo-ments must involve the antiferromagnetic RKKY inducedinter-site exchange coupling, which competes with theferromagnetic ones. As we demonstrate below, this com-petition results in several QCPs in multi-impurity modelsof the second kind.Nozieres [14, 15] and others [65] already suggested thatonly the fraction T K ρ (0) of the conduction electrons cancontribute to the Kondo screening in the PAM, T K beingthe single-impurity Kondo temperature and ρ (0) beingthe conduction electron density of states at the chemi-cal potential. Therefore, there are not enough conduc-tion electrons available for ensuring the Kondo screeningof all local f -moments by independent Kondo screeningmechanisms.The singlet ground state formation in a heavy Fermiliquid must be based on a different mechanism then sim-ply extending the single-impurity Kondo effect to peri-odic structures. Although the PAM is mapped onto aneffective single-site problem [66–70] embedded into a lat-tice self-consistency condition in the context of the dy-namical mean field theory (DMFT) [71], indicating a sim-ple connection between the single-ion Kondo effect and the Kondo lattice problem, Pruschke and collaborators[70] interpreted the occurring chemical potential depen-dent reduction of the effective conduction electron den-sity of states in the effective single-site problem in termsof Nozieres’ exhaustion scenario.Based on our low-energy mapping presented below, weprovide a different perspective on the heavy FL forma-tion in the PAM and Kondo lattice models (KLM). Itreplaces the phenomenological exhaustion scenario witha rigorous mathematical criterion and connects the lo-cal Kondo screening and magnetic ordering within theDMFT approach to the mechanism [72] known from theHubbard model. C. Low-energy effective multi-impurity model
Since an exact solution of complex multi-impurity cor-related electron systems is not known in most of theinteresting cases, when the number of impurities ex-ceeds N f >
2, the challenge is to find an appropriateapproximation to nevertheless extract the relevant low-temperature physics. We propose a mapping onto aneffective low-energy model which can be used to analyzethe emergence of free local moments in a variety of differ-ent situations, and allows to understand their screeningas well as the potential magnetic ordering.The effect of the host conduction band onto the dy-namics of the correlated lattice sites is determined bythe hybridization function matrix [73],∆ lm ( z ) = 1 N c X ~k V ∗ ~k,l V ~k,m e i~k ( ~R l − ~R m ) z − ǫ ~k , (6)derived from Eq. (3).The exact real space multi-impurity Green’s functionmatrix of the dimension N f × N f , in the absence of theCoulomb interaction, U l = 0, is given by the matrix G ( z ) = [ z − E − ∆ ( z )] − , (7)where the matrix E contains the single-particle energiesof the localized orbitals [74], and the matrix elements ofthe self-energy matrix ∆ ( z ) are given in Eq. (6).The energy dependence of ∆ lm ( z ) can be neglected inthe wide band limit, V l /D →
0. Its effect onto the lo-cal impurity dynamics is mainly determined by the com-plex matrix elements ∆ lm ( − i + ) for a ρ ( ω ) that is al-most constant on the relevant low-energy window. Wealso have to be careful with the distance dependency ofthe off-diagonal matrix elements ∆ lm ( z ). The definition(6) reveals that the larger the distance, the more pro-nounced the frequency oscillation of ∆ lm ( z ) are close toFermi energy. The error of the approximation is esti-mated by the first derivative of the imaginary part of∆ lm ( z ), d Γ lm ( ω ) /dω [32]. While the oscillations are verypronounced for an isotropic dispersion of the conductionelectrons in 1d, resulting in significant corrections in 1d,the derivative becomes R independent in 3d – see Eq.(61) in Ref. [32]. The proposed approximation is validin the regime d Γ lm ( ω ) /dω → V l /D →
0. However, as we demonstrate lateron, the fixed point of the full model in general is correctlyincorporated in the effective model, even for finite
V /D .We divide the complex matrix element ∆ lm ( − i + )into its real and imaginary part: ∆ lm ( − i + ) =Re∆ lm ( − i + ) + i Γ lm . We absorb the effective inter-orbital hopping matrix elements Re∆ lm ( − i + ) into theenergy matrix E → E + Re ∆ ( − i + ). If we are onlyinterested in the dynamics of the multi-impurity clusterdegrees of freedom, the problem can be mapped ontoan effective problem where the charge fluctuation matrixΓ lm is generated by an fictions set of conduction bands.For that purpose, we diagonalize the Hermitian matrixΓ lm , Γ = U h Γ d U (8)where the N f eigenvalues Γ dn = π ¯ V n ρ (0) are interpretedas coupling of the new orbital n to the n-th new effec-tive conduction band with the same DOS as the originalconduction band DOS, and the hybridization strength isgiven by ¯ V n . This is justified since we are only interestedin the different fixed point structure of the model andnot in the accurate calculation of the low temperaturecrossover scale. Its precise number is also determined bythe high-energy degrees of freedom [39, 50, 75]. Then theGreen’s function is approximated by G ( z ) ≈ U [ z − E ′ − i Γ d ] − U h , (9)where the energy matrix E ′ E ′ = U [ E + Re ∆ ( − i + )] U h (10)contains diagonal and hopping terms between all corre-lated impurity orbitals in the new eigenbase diagonalizing Γ . The approximation is limited to a range of frequencies z for which ∆ lm ( z ) ≈ ∆ lm ( si + ) with s = sign(Im z ).Consequently, the same low-frequency single-particleGreen’s function matrix is generated by the effectivesingle-particle Hamiltonian H ′ sp = H cl + H deloc (11)in the new eigenbase of Γ in the limit V l /D →
0. Thecluster part of the mapped Hamiltonian H ′ sp , H cl = X m,l E ′ lm f † l f m , (12)defines the single-particle Hamiltonian of the correlatedorbitals in the new basis that have acquired additionalorbital hopping terms mediated by the conduction bandof the host. The second part, H deloc = N f X n =1 X ~k ǫ ~k c † ~k,n c ~k,n (13)+ N f X n =1 X ~k (cid:18) ¯ V n √ N c c † ~k,n f n + h.c. (cid:19) , includes the new N f effective conduction band degreesand the flavor diagonal coupling to the cluster orbitalsfor each conduction band flavor n . Note, however, thatthe total number of particles in each flavor n is in generalnot conserved, since this operator does not commute withthe single-particle cluster Hamiltonian H cl .Whereas the hybridizations ¯ V n of the mapped modelare exclusively determined by the Fermi-surface, all highenergy conduction band states contribute to the dynam-ics of the cluster H cl via the energy matrix elements E ′ lm .The real part of the complex hybridization function, andthe effective hopping elements t eff lm respectively, can bededuced via a Hilbert transformation t eff lm = Re∆ lm ( − i + ) = 1 π Z ∞−∞ dǫ Γ lm ( ǫ ) ǫ (14)that incorporates the whole energy dependence of thecoupling functions. These hopping elements generate theantiferromagnetic part of the RKKY interaction and si-multaneously lead to destruction of the QCP in the twoimpurity limit [11, 12, 32] since it is a relevant perturba-tion of the fixed point [11].The correlated MIAM is recovered after the localCoulomb matrix elements in H corr , Eq. (2), has also beenrotated into the new orbital basis as well, and addedto the single-particle Hamiltonian H ′ sp . This leads to acomplicated, coupled multi-impurity problem that stillcontains the full spatial correlations in contrary to a lo-cal approximation in real space that is employed by theDMFT.This mapping generates N f fictitious conduction bandslabeled with the index n and the corresponding, orthogo-nal single particle orbitals, augmented by new orbital en-ergies and an inter-orbital hopping, both included in thematrix elements E ′ lm . As a side product of this mapping,we have separated the AF part of the RKKY interactionthat is generated by the inter-site matrix elements of E ′ lm from the Kondo screening channels. Since the hybridiza-tion strengths Γ n are in general all different, multi-stagescreening of local moments is found in such situations.Furthermore, the differences generate the FM part of theRKKY interaction [31, 32]. In addition we found thatfor large N f , only a few Γ n are different from zero, there-fore, the number of Kondo screening channels is typicallymuch smaller than N f .This mapping was previously investigated [32] in thetwo-impurity Anderson model ( N f = 2) where the twoconduction bands represent states with even and withodd parity. In this case, one can either use the full energydependency of the even-parity and the odd-parity band[11, 12, 76, 77] in an numerical renormalization group(NRG) [39] calculation, or investigate the mapped Hamil-tonian (11) with a particle-hole symmetric band densityof states. Both Hamiltonians, the original MIAM, aswell as the mapped Hamiltonian, produced the same RGfixed points for a featureless initial ρ ( ω ), and the samespin-spin correlation functions in the wide band limit, es-tablishing the quality of the mapping for N f = 2. It wasshown that the particle-hole asymmetry in the even andodd conduction band dynamically generates an effectivehopping between the two local orbitals, which is respon-sible for the AF part of the RKKY interaction via anexchange mechanism once U l >
0. The FM part of theRKKY interaction is generated by the imbalance betweenthe two eigenvalues of Γ . D. Classification of the multi-impurity problem bythe rank( Γ ) Having an effective low-energy multi-impurity modelat hand, we can now rigorously define different classes ofMIAMs, as well as distinguish between a multi-impuritymodel of the first kind versus one of the second kind.Mathematically, the hybridization matrix Γ has exactly N f eigenvalues if its rank is equal to the number of im-purity orbitals, i. e. rank( Γ ) = N f . Since the originalmodel only contains a single conduction electron band,the phase correlations between different lattice sites en-coded in Eq. (6), however, are responsible that the rankof the matrix is often less than the number of impurity or-bitals: rank( Γ ) < N f . Therefore, we use the rank( Γ ) toclassify the MIAM into two categories: A multi-impurityproblem of first kind requires that rank( Γ ) = N f , whilerank( Γ ) < N f defines a multi-impurity problem of sec-ond kind.The multi-impurity problem of first kind is an exam-ple for a compensated multi-channel Kondo problem [63]:there are always enough conduction band channels avail-able for a complete screening of all local moments via amulti-stage Kondo effect [63]. When reducing the tem-perature of the system, the details of the eigenvalues Γ dn of Γ define a cascade of low-energy scales at which the lo-cal moments are quenched by 1 / E ′ lm is re-sponsible for generating an effective low-energy Heisen-berg model, representing the AF part of the RKKY inter-action, the pre-quenching of the moments via the effec-tive Heisenberg couplings is not needed to obtain a sin-glet ground state. Consequently, ferromagnetic exchangecouplings between the local moments are irrelevant withrespect to the singlet ground state for models of the firstkind. However, interesting physics can arises in problemsthat contain magnetically frustrated systems requiring atleast three impurities [33–35, 37, 38]. Our mapping pro-vides an ideal tool to investigate which physical condi-tion the original model must fulfill in order to reach thecritical parameter regimes reported for the trimer Kondomodels [33–38].The two-impurity Anderson (TIAM) or Kondo modelis a typical representative of a multi-impurity problem offirst kind where always a singlet ground state is gener-ated - with the exception of peculiar geometric conditions[32, 78, 79] where rank( Γ ) = 1 is found. Although theinterest in the TIAM was driven by the Doniach scenario,the originally reported quantum phase transition [31] be- tween a two-stage Kondo singlet and an RKKY inducedsinglet turned out to be an artifact of the approxima-tion. This model shows a crossover between both phasesaccompanied by a continuous variation of the scatteringphase [11, 12] which is also included in our mapped model[32]. The QPT is destroyed by the real part of ∆ ( − iδ )inducing a hopping term in the cluster that is a relevantperturbation in the vicinity of the QPT [11, 12, 32].The TIAM is an ideal system to explicitly understandthe origin of the rank reduction in our effective model,since the original mapping by Jones and Varma alwaysleads to a coupling to two conduction bands. Followingthe arguments of Refs. [32, 79], or inspecting the imag-inary part of ∆ lm ( z ), Eq. (6), in the even or odd par-ity basis for certain dispersions ǫ ~k and relative distances ~R l − ~R m between the impurities, yields a vanishing of theenergy dependent coupling function Γ( ω ) at the chemi-cal potential of a power-law form | ω | α , where α >
1. Forsuch an exponent of pseudo-gap coupling functions thelocal moment fixed point has been proven to be stable[80–82] in the RG flow. Therefore, the approximationmade in the effective low-energy model (11), by neglect-ing the full energy dependency of the bands, is fully jus-tified since the low-energy fixed point remains unaltered.The rank of the coupling function matrix Γ is a simplemeasure to identify the number of independent effectiveconduction electron channels that can be potentially usedfor the screening of local moments by the Kondo effect.A interesting consequence arises for large N f , for in-stance in the PAM where N f = N L . Let us consider avery large but finite system with periodic boundary con-ditions. In this case, we know that the problem can bediagonalized in ~k space: The new multi-impurity orbitalsare labeled also by the quantum number ~k , and acquire avery complicated non-local Coulomb matrix. The single-particle matrix ∆ ( z ), however, must be diagonal in ~k ,and the matrix elements take the very simple form∆ ~k ( z ) = | V ~k | z − ǫ ~k , (15)which is the well known self-energy of the f -lattice Greenfunction. As a consequence, only the ~k -values for which ǫ ~k = 0 holds, yield a finite Γ ~k in the mapped model.Therefore, rank( Γ ) ≪ N f for the PAM, and the num-ber of available screening channels is related to the sizeof the Fermi surface and not by the number of corre-lated orbitals. We can conclude that in one-dimensionrank( Γ ) ≤ Γ ) allows a much more pre-cise definition of the phenomenological exhaustion prin-ciple: for a Kondo screening in the MIAM there are onlyrank( Γ ) screening channels available. Obviously, this def-inition is only governed by the single-particle properties,introduced by the arrangement of the impurities, the un-derlying lattice and the host dispersion ǫ ~k . This math-ematically precise definition, however, is able to replacethe phenomenological notion of a fraction ρ (0) T K of elec-trons contributing to the Kondo screening, which requiresthe definition of T K although T K became a questionablequantity in MIAM.This finding is a strong indicator that the singletground state in the PAM is caused by a different mech-anism: it is driven by the hopping matrix elements E ′ lm delocalizing the local impurity electrons within the f -impurity subsystem and not by N f independent conduc-tion electron channels, as already conjectured by Grewe[13] more then 30 years ago. For this second kind of MI-AMs, the formation of large spins due to ferromagneticexchange couplings competes with the self-screening ofthe correlated electrons, and leads to several QCPs inthe phase diagram of such models. Interesting physicsalso arises from the competition between self-screeningof the correlated impurity cluster and magnetically frus-tration due to long range hopping matrix elements E ′ lm in finite dimensions. By inspecting Eq. (6), one can con-clude that E ′ lm decays rather rapidly in higher spatial di-mensions destroying the physics of magnetic frustrationin the limit d → ∞ , in accordance with the arguments ofMetzner and Vollhardt [83], Brandt and Mielsch [84] aswell as M¨uller-Hartmann [85].
1. The rank( Γ ) for finite impurity-cluster in variousdimensions In the PAM the number of k-points on the Fermi sur-face of the host material provides the upper limit for thenumber the available screening channels, independent ofthe structure and the dimension of the underlying lat-tice. Since the number of decoupled f -orbitals in thePAM must continuously develop out of the MIAM witha finite number N f of correlated impurities, we study thereduction of rank( Γ ) for finite impurity clusters with dif-ferent geometrys and in different dimensions. For thispurpose we consider a simple cubic lattice with nearestneighbor hopping t such that the dispersion ǫ ~k in d di-mensions reads ǫ ~k = − t P di cos( k i a ) + ǫ c . In the follow-ing we concentrate on dense impurity arrays where all theimpurities are placed next to each other. Dilute impurityconfigurations can always be deduced from a dense array,by shifting the on-site energy ǫ fi of the depleted sites toinfinity. Therefore, the rank( Γ ) of the dense array servesas an upper limit for any depleted configuration that canbe deduced from the dens case.In one dimension, the Fermi surface consist of twosingle points which determine the rank of the charge-fluctuation matrix for the PAM: rank( Γ ) = 2. Conse-quently for any finite number N f of correlated impuritieswe can conclude rank( Γ ) ≤ N cf = 3. Every MIAMin 1d with N f ≥ N cf belongs to the MIAM of second kindand thus exhibits QCPs due to FM correlations betweenthe local moments in its parameter space.In higher dimensions the Fermi surface itself becomesa continuum in the thermodynamic limit and an argu- FIG. 2. MIAM with 2d simple cubic lattice, containing un-correlated lattice orbitals (green) and correlated impurities(blue). The individual panels depict different geometries ofthe impurity configuration leading to a different number ofdecoupled orbitals N free = N f − rank( Γ ). (a) N free = 0, (b) N free = 1, (c) N free = 1 and (d) N free = 2. mentation analogue to the 1d case is not possible. Hencewe focus on some explicit configurations in 2d which areschematically depicted in Fig. 2If the impurities are placed in line along the x -directionas schematically depicted in Fig. 2 (a), we can diagonalizethe charge-fluctuation matrix in the limit N f → ∞ via a1d Fourier transformationΓ k x ∝ X k y | V ~k | δ ( ǫ ~k ) . (16)For a half filled conduction band, ǫ c = 0, one can al-ways find a k y such that ~k = ( k x , k y ) T belongs to theFermi surface and, consequently, Γ k x = 0 which implies aMIAM of the first kind: rank( Γ ) = N f . Deviations fromhalf-filling, ǫ c = 0, lead to a small number of k x -pointsfor which no k y can be found such that ~k = ( k x , k y ) T belongs to the Fermi surface. In this case the MIAMwith N f → ∞ belongs to the second kind, however, thenumber of decoupled orbitals N free = N f − rank( Γ ) re-mains small and for N f ≤
50 a numerical evaluationyields rank( Γ ) = N f for various fillings.For a finite number of N f = 5 impurities, which arearranged as depicted in panel Fig. 2(b) we can analyt-ically calculate rank( Γ ) using the irreducible represen-tation of the C point group. In this basis one obtainsthree 1d subspaces with Γ n = 0, for a general filling ǫ c ,and one 2d subspace. The 2d subspace contains the cor-related orbital in the center of the impurity array f c andthe even combination f e = P i f i,o , where f i,o denotesthe annihilation operator of the outer impurities. Thecharge-fluctuation matrix of this 2d subspace for arbi-trary fillings ǫ c reads Γ even = Γ outer e Γ o/c e Γ o/c e Γ center e ! = Γ (cid:18) ( ǫ c /D ) ǫ c /Dǫ c /D (cid:19) , (17)and exhibits an incomplete rank for every ǫ c due todet( Γ even ) = 0. We can conclude rank( Γ ) = 4 = N f − N f impurities on a 2dsimple cubic lattice, the charge-fluctuation matrix Γ canbe numerically evaluated using Eq. (6). For the con-figuration in panel (c) of Fig. 2, for instance, we ob-tain rank( Γ ) = 8 = N f −
1, whereas the evaluation ofrank( Γ ), for an arrangement as depicted in Fig. 2 (d),yields rank( Γ ) = 6 = N f −
2. The analytical and numer-ical evaluation of the rank of the charge-fluctuation ma-trix Γ for a simple cubic lattice in 2d, as well as in 3d, in-dicates, that the number of available screening-channelsis proportional to the number of the outermost impuri-ties of a certain arrangement. This finding is compatiblewith the fact, that rank( Γ ) for the PAM is limited tothe number of ~k -points on the Fermi surface of the hostmaterial, which in general is proportional to N d − .For STM experiments, impurity cluster on a 2d sur-face of a 3d crystal are of particular interest. In order toqualitatively study such situations we used the exact 2dsurface Greens function G ( ~k || , ω ), with ~k || = ( k x , k y ) T ,of a semi-infinite 3d simple cubic lattice, which can befound in [86, 87], to construct the complex hybridizationmatrix ∆ ( z ). If the number N f of impurities is small,we found a MIAM of the first kind in general. Con-figurations which belong to the second kind of MIAMsin the pure 2d case, however, exhibit a clear hierarchyof hybridizations Γ n in the mapped model. Hence, dueto the finite temperature in experiments, the fully spin-compensated ground state may not be reached since thesmallest Kondo temperature is exponential suppressed.For a large number of impurities arranged in a densecluster one will continuously reach the limit of a fullycovered 2d surface. In that case we can diagonalize thecharge-fluctuation matrix Γ via a 2d Fourier transforma-tion which, according to [86, 87], yieldsΓ ~k || ∝ s − (cid:18) ǫ ~k || / t (cid:19) if (cid:12)(cid:12)(cid:12) ǫ ~k || / t (cid:12)(cid:12)(cid:12) ≤
10 else , (18)and, consequently, reveals a MIAM of the second kind. E. Constructing effective cluster models in thelocal moment regime
Having an effective multi-impurity model at hand, wecan use the results of Sec. II C to propose a two stepprocess in order to gain some physical insight in the low-energy properties of the original model. In a first step, we
FIG. 3. The diluted periodic Anderson model in 1d: the localimpurities are only connected to the B sublattice set the couplings ¯ V n to the effective conduction bands tozero, and focus on the decoupled cluster dynamics. Af-ter understanding the ground state and the elementaryexcitations within the cluster, we couple the cluster tothe neglected conduction bands. Such a procedure im-plies a certain hierarchy of energy scales: the Coulombrepulsion, being the largest energy scale causes a localmoment formation, dividing the cluster Hilbert space inirreducible subspaces of the total spin. The hopping ma-trix defines the intermediate energy scale selecting theground state multiplet of the cluster. In the last step,the effective local moment fixed point of the decoupledcluster becomes unstable due to the coupling to the ne-glected conduction bands.For this energy hierarchy one can employ a two stepSchrieffer-Wolff type [88] transformation. In a first step,such a transformation is applied to the decoupled cluster.This leads to a finite size t-J model for large N f as usedin the context of the high temperature superconductors[5]. Depending on the particle-hole asymmetry, a purespin model might emerge, favoring locally antiferromag-netically aligned spins that might order for N f → ∞ , or amore complicated model with two and three site interac-tions. In a second Schrieffer-Wolff type [88] transforma-tion, an effective Kondo coupling is obtained between theground state multiplets and the now included coupling tothe previously neglected conduction bands.In the SIAM [39, 46, 47], the system flows to samestrong coupling fixed point in case of a dominating cou-pling to conduction electrons but without a clear sig-nature of of the local moment fixed point. Therefore,the energy hierarchy outlined above is not essential ina full NRG and only helps shaping our physical intu-ition in some limited cases. Hence, our low-energy MIAMalso contains the correct physics for the cases in whichthe hybridization strengths ¯Γ n dominate over the mag-netic exchange terms that would be generated by the firstSchrieffer-Wolff type transformation. In this case the lo-cal moments are starting to be partially screened beforethe interaction between the magnetic moments becomesrelevant.0 F. Applying the mapping to some cases discussedin the literature
1. The dilute Kondo lattice model in 1d
An interesting version of a diluted Kondo lattice modelin 1d was investigated by Potthoff and collaborators [42,43] at half-filling. The correlated electron sites are onlycoupled to the B-sublattice of the bi-partite lattice asdepicted in Fig. 3. The authors report that the groundstate is ferromagnetic, and the total spin S = ( N f − / a forms a bi-bipartite lattice. A simple nearestneighbor tight binding band structure, ǫ k = − t cos( ka ),was considered by Potthoff and collaborators [42, 43, 89,90]. Starting from a half-filled ( k F = ± π/ a ) conductionband, the real and imaginary parts of ∆ lm ( − i + ) aregiven byIm∆ lm ( − i + ) = πV l V m ρ (0) cos (cid:18) π R lm a (cid:19) (19a)Re∆ lm ( − i + ) = V l V m Z D − D dǫ ρ ( ǫ ) cos (cid:0) cos − (cid:0) ǫD (cid:1) R lm a (cid:1) ǫ . (19b)If the impurities are placed on the same (different) sub-lattice, i.e. R lm = 2 na ( R lm = [2 n + 1] a ) with n ∈ Z , thereal (imaginary) part of ∆ lm ( − i + ) vanishes. For V l = V we obtain∆ lm ( − i + ) = ( ± i Γ same sublatticeRe∆ lm ( − i + ) different sublattice(20)where Γ = πV ρ (0). Then, the hybridization functionmatrix of the correlated lattice sites at the sublattice B,as depicted in Fig. 3, is given by a purely imaginary ma-trix ∆ = i Γ +1 − · · ·− − · · · +1 − · · · ... ... ... . . . (21)at zero frequency. The matrix is finite dimensional fora finite lattice of size N c = 2 N f with rank( − i ∆ ) = 1,such that only a single eigenvalue ¯Γ = N f Γ is differ-ent from zero. Only one of the N f new orbitals cou-ples to an effective conduction electron band, and in realspace a 1 /N f fraction of the local moments on each lat-tice site is screened by the flow to the strong couplingfixed point. The other N f − ∆ ] = 0. Since the asymmetry of the coupling constants is respon-sible for the FM part of the RKKY interaction [31, 32],those remaining N f − V ≫ V l , l = 0. Thenwe can solve the problem in two steps: solve a single-impurity problem by setting V l = 0 , l = 0, and once thelow-energy fixed point of that problem is reached, switchon the other couplings V l . The diagonal component ofthe conduction electron Green’s function in real space, G ii ( z ) = G ii ( z ) + V G i ( z ) G U imp ( z ) G i ( z ) , (22)is obtained from the exact equation of motion and deter-mines the local density of states prior to switching on V l . G ij ( z ) denotes the free propagator from R i to R j G ij ( z ) = 1 N X k e ik ( R i − R j ) z − ǫ k = Z D − D ρ ( ω ) cos (cid:8) ( R i − R j ) cos − (cid:2) ωD (cid:3)(cid:9) z − ǫ (23)and V G U imp ( z ) the t-matrix of the impurity located atthe origin, calculated for a finite U . For a particle-holesymmetric band, the real part of G ij ( z ) essentially van-ishes for ( R i − R j ) = 2 na , and n ∈ Z , at low frequencies.If we substitute a Lorentzian approximation for G U =0imp ( z )(Kondo effect), we obtain the approximate solution ρ R i ( ω ) ≈ ˜ ρ R i ( ω ) = ρ ( ω ) − h ρ ( ω ) cos n R i cos − h ωD ioi π V ρ (0) ω + [ πV ρ (0)] (24)for the local conduction electron spectral function at site R i . A careful analysis of this DOS reveals a pseudo-gapformation of the spectrum at all other correlated impu-rity sites: the larger the distance, the faster the DOS os-cillations in energy space, the smaller the energy intervalof the pseudo-gap. Since the pseudo-gap always vanishesquadratically, Γ i ( ω ) ∝ | ω | in this energy window, a lo-cal magnetic moment of another impurity coupled to theconduction electrons at site R i remains unscreened in thelimit T → V l , in favor of a single dominating hy-bridization, provides a real space interpretation of ourfinding that rank( Γ ) = 1 in the 1d dilute PAM. The de-localized orbital coupling to the single effective band for1 V l = V is adiabatically deformed to the localized orbitalat the origin for V l ≪ V ( l = 0), which does not alterthe rank of the matrix. In our approach, the real spacenature of this orbital is encoded in the eigenvector of Γ corresponding to the single finite eigenvalue.In an approach that includes the full energy depen-dency of the conduction bands in an effective N f bandmodel, we obtain N f − ǫ k and the topology of the impurity locations. Conse-quently, the fixed point of the full model is correctly cap-tured by the effective low-energy Hamiltonian, even forlarge V /D apart from the wide band limit, contrary to lo-cal quantities. As demonstrated by DMRG calculations[42, 43], the size of the magnetic moment of the groundstate is independent from the strengh of
V /D . However,whereas it posses mainly f -character in the wide bandlimit V /D →
0, it continuously shifts into the host, andbecomes a pure conduction band quantity in the limit
V /D → ∞ . This flow of the magnetic moment, from f -to c -character, obviously is absent in the effective low-energy description for the wide band limit.
2. Limit of the one-electron Kondo-lattice model
Extending the consideration to the full PAM in 1dat half-filling, the problem can be interpreted as cou-pling the two dilute PAM problems of each sublatticevia the hopping matrix elements define in Eq. (19b). Oneach sublattice, one spin is screened while the remain-ing N f / − S tot =( N f − / µ → − td or ǫ c → td , whilekeeping the correlated site singly occupied, where d is thedimension of the simple cubic lattice. For | ~k F | →
0, weobtain Γ lm = Γ and as a consequence again rank( Γ ) = 1.The same holds for ~k F → π (1 , · · · , T . Therefore, oneeffective moment is screened, and our mapping predictsa FM ground state of S tot = ( N f − / J → ∞ of theKondo lattice model, where a ground state formed by lo-cal singlets is expected, is a singular point [44] and notaccessible within the PAM. While the exact form of the ground state and the ground state energy still require adetailed calculations, the rank of Γ provides already a ba-sic understanding of the elementary magnetic propertiesof the model in the limit of a vanishing band filling.
3. Mott transition in the periodic Anderson model withnearest neighbor hybridization
As a third example we consider the periodic Andersonmodel with nearest neighbor hybridization [45]. The real-space matrix ∆ ( z ) becomes diagonal in ~k -space,∆ ~k ( z ) = | V | t ǫ ~k z − ǫ ~k . (25)Clearly, the imaginary part of ∆ ~k ( ω − iδ ) vanishes for ahalf-filled conduction band for all ~k , reducing rank( Γ ) =0: no effective conduction band couples to the impurityorbitals, and one is left with a decoupled infinite sizecluster of f -orbitals.The real part of ∆ ~k ( z ), takes the form˜ t ~k = − | V | t ǫ ~k , (26)leading to a renormalized band dispersion for U = 0 ǫ f~k = ǫ f − | V | t ǫ ~k (27)in ~k -space. Our approximation maps the PAM with near-est neighbor hybridization onto an effective single-bandHubbard model. The band dispersion of this decoupled f -electron subsystem is identical to those of the host con-duction electron but rescaled by the factor ( V /t ) and adifferent sign.Without any further calculations, we can concludethat this model must undergo a Mott-Hubbard metal-insulator transition at two different critical values U fc , U fc for T → U fc are related to the established re-sults for the Hubbard model by the scaling factor V /t .This prediction of our mapping perfectly agrees withthe elaborate DMFT calculation by Held and Bulla [45].They found (i) a Mott-Hubbard metal-insulator transi-tion in their full DMFT calculation and (ii) a scaling of U fc = ( V /t ) U Hub c for small V . The scaling factor ismodified for larger hybridization strength V , exactly asdiscussed in Sec. II C: corrections must be included forthe deviations from the wide-band limit. This demon-strates the power and the potential of our mapping forthe qualitative understanding of such correlated latticemodels.2 III. NUMERICAL RENORMALIZATIONGROUP RESULTS FOR MULTI-IMPURITYPROBLEMSA. Emerging trimer with three conduction electronbands
In the last decades the role of magnetic frustration forHF has been discussed in the context of strongly cor-related multi-impurity models [4, 33–38]. Since it wasrealized that the two-impurity problem lacks the com-plexity and the reported QCP appears to be an artifactof the approximation [11, 12] made in the originally ap-proach, nowadays the focus lies on models with threelocal moments coupled to an arbitrary number of conduc-tion bands [33–38]. These papers in particular focus onthe emergence of a frustration induced NFL fixed pointthat might be related to QCPs found in bulk materials.In this section we review these kind of models from theperspective of our effective low-energy mapping.In case of three identical correlated orbitals, the struc-ture of the effective low-energy model reduces to fourfree parameters: The diagonal elements of ∆ ( − i + ) areequal for all impurities, whereas the off-diagonal ele-ments ∆ off ij ( − i + ) depend on the geometric arrangementof the impurities and the structure, as well as the fill-ing, of the underlying lattice of the crystal. For Γ off ij =Im∆ off ij ( − i + ) = 0 each of the correlated orbitals couplesto its own, independent conduction band, as studied in[35]. In general, however, the impurities are coupled tothe same conduction band, which implies Γ off ij = 0, suchthat a precursive diagonalization of Γ is necessary in or-der to obtain an ”independent-bath” description of themodel.For a generic setup of the three impurity problem ina metallic host, the Hamiltonian does not preserve thesymmetry of the C point group, and, consequently, atleast either Γ off12 = Γ off23 or Re∆ off12 ( − i + ) = t eff12 = t eff23 . Forsuch a situation a frustration induced NFL fixed pointis unlikely to be found, since either the Kondo-scale orthe RKKY-scale induces an imbalance that prevents apossible frustration.In some special setups, however, the Hamiltonian maypreserve the symmetry of the C point group, as studiedin Ref. [33, 34], where the Kondo-impurities are placedin a crystal with an isotrop dispersion relation in an ar-rangement of an equilateral triangle with impurity sepa-ration ∆ R . In this case, the authors found three differentfixed points.If the ferromagnetic RKKY interaction dominates, thelocal moment that forms at intermediate temperature isscreened by the three independent effective conductionbands, leading to a FL with spin-singlet ground state.This corresponds to the trivial case of individual Kondo-screening of the three local moments for ∆ R → ∞ andis referred to as the independent Kondo FP. In case ofa dominating antiferromagnetic RKKY interaction, theauthors of Refs. [33, 34] still differentiate between two frustrated scenarios, which they called ”frustrated Kondoregime” and ”isospin Kondo-regime”, both characterizedby NFL FP (for further explanation see Ref. [33] and[34]). Whereas the isospin Kondo-regime was found tobe unstable against weak PH asymmetry, the frustratedKondo regime is robust against moderate PH asymmetry.In order to study the model with a C symmetry withinour effective low-energy mapping, we applied the NRGwith a discretization of λ = 4 and kept N s = 3000states after each diagonalization in the NRG procedure.Note, that even if N s = 3000 states might be insufficientto calculate thermodynamic quantities within a three-channel NRG calculation, the authors of Ref. [33] onlykept N s = 1200 states in their NRG calculations, whichwas sufficient to reproduce a conformal field theoreticaldescription [34] of the model.Due to the C symmetry, two independent parametersdetermine the influence of the original host conductionband onto the dynamics of the correlated orbitals in theeffective low-energy model, Γ off = Γ off12 = Γ off23 and t eff = t eff12 = t eff23 . In this section we focus on the parameterregime − . < Γ off / Γ <
1, such that rank( Γ ) = 3 issatisfied, and we are always studying a MIAM of thefirst kind.The case of dominating ferromagnetic RKKY inter-action, or ∆ R → ∞ respectively, corresponds to small t eff → off .For dominating AF RKKY interactions, correspondingto an appropriate value of t eff , we can still differentiatebetween two situations: Applying C point group prop-erties, two of the three eigenvalues of Γ are identical.These are associated with the two helical states whichare complex conjugated to each other. For Γ off / Γ < off / Γ >
0, thesingle eigenvalue dominates. Γ off / Γ = 0 implies an inde-pendent conduction band for each impurity in real spaceand, consequently, to three identical eigenvalues Γ n = Γ .In order to artificially restore PH symmetry, which isnecessary to reproduce both NFL fixed points found inRef. [33, 34], we replace the effective tunneling elements t eff by an effective Heisenberg exchange interaction J =4( t eff ) /U between the correlated orbitals X σ X ij,i = j t eff f † iσ f jσ → X ij,i = j J ~S i ~S j . (28)This incorporates the correct AF part of the RKKY in-teraction but removes the PH symmetry breaking termby hand. For this PH symmetric setting, and J/T K ≫ off / Γ ≤ off / Γ >
0. Whereasthe isospin Kondo FP is unstable against small devia-tions from the PH symmetric point, ǫ f → ǫ f ± δǫ , thefrustrated Kondo FP remains stable in accordance withRefs. [33, 34]. However, our NRG calculations reveal,that even the frustrated Kondo FP is unstable when a3 FIG. 4. Phase diagram for the low-energy model with C symmetry as function of the tunneling element t eff betweenthe correlated orbitals and Γ off / Γ = 0 . U/ Γ = 20, ǫ f / Γ = − D/ Γ = 10. Small values of t eff result in a FM or weakAF exchange interaction and a FL. Intermediate t eff c, < t eff
18, theAF part of the RKKY interaction dominates, leading tothe frustrated Kondo regime with irrational degeneracy g of the ground state, g = [1 / √ . , as reportedin [34].While further increasing t eff , we observe a secondcritical value t eff c, / Γ ≈ .
41 at which the NFL fixedpoint becomes unstable: For t eff > t eff c, , the systemflows to the unstable frustrated Kondo fixed point atintermediate temperatures, but crosses over to a sta-ble FL below the low-energy scale T (purple linepointsin Fig. 4), which is defined at the point where the en-tropy reaches the midpoint between both fixed points, S imp ( T ) = 1 / S FLimp + S NFLimp ). This new crossover scaleis exponentially suppressed when approaching the QCPfor ( t eff − t eff c, ) → + .In order to find a finite interval of t eff in which the NFLfixed point is stable, the total RKKY coupling K RKKY ,comprising the FM and AF contributions K RKKY = K FMRKKY + K AFRKKY needs to fulfill two conditions: (i) theAF part of the RKKY interaction has to dominate overthe FM one, K AFRKKY > K
FMRKKY , and, (ii) K RKKY needsto be larger than the single-ion Kondo temperature butmust not exceed an upper critical value T K < K RKKY < K c RKKY , (29)since an increasing K AFRKKY is associated with increasingpotential scattering destroying the NFL fixed point. Asa result of the upper bound in Eq. (29), the NFL fixedpoint is absent once T K > K c RKKY .We repeated the NRG calculations for two other ra-tios of Γ off / Γ as well and find the same sequence oflow-energy fixed points. We summarized the two crit-ical t eff c, / for three different values of Γ off / Γ and twodifferent Coulomb interactions U/ Γ , for PH symmetricimpurities, i.e. ǫ f = − U/ off / Γ , the larger is the interval ( t eff c, − t eff c, ) of the NFLregime as apparent for U/ Γ = 20. Since a large Γ off in-creases the FM RKKY interaction, a larger t eff > t eff c, isnecessary in order to reach the stable frustrated Kondoregime.In case of U/ Γ = 10, the single-ion Kondo tempera-ture is too large in order to fulfill Eq. (29), and the NFLfixed point is absent for all three values of Γ off / Γ listed4 FIG. 5. Effective tunneling element t eff = Re∆ off ( − i0 + )(black) and Γ off = Im∆ off ( − i0 + ) (light blue) as function ofthe dimensionless distance Rk F for an isotropic linear disper-sion ǫ ~k in (a) 2d and (b) 3d. in table I. Consequently, a small Kondo temperature incombination with large Γ off and t eff stabilizes the NFLfixed point over a large region in the parameter space.However, Γ off and t eff can not be chosen independentlyin real materials, since they result from the imaginaryand real part of the same complex hybridization matrix ∆ ( − i + ). In Fig. 5, we plotted Γ off (light blue) and t eff (black) in (a) 2d and in (b) 3d, as function of the dimen-sionless distance Rk F , using a isotropic linear dispersion ǫ ~k for host conduction band. Due to the typical phaseshift between Γ off and t eff , large values of t eff in generalcorrespond to small values of Γ off , destabilizing the NFLfixed point.In conclusion, our low-energy mapping demonstrates,that it indeed is possible to realize the NFL fixed pointof the frustrated Kondo regime found in Refs. [33, 34].Nevertheless, aside from the C symmetric setup, this re-quires a substantial small single-ion Kondo temperatureand a small impurity separation Rk F ≈ off / Γ > t eff c, < t eff < t eff c, . B. Multi-impurity problems connected to a 1d host
In this section we present the results for several MI-AMs of the second kind by applying the NRG to theeffective low-energy models. We focus on the low-energyfixed points, the crossover temperatures and spin-spincorrelation function for N f = 3 , N f = 5 as func-tion of the band filling, and identify a series of Kosterlitz-Thouless type QCPs that are associated with a change ofthe fixed point degeneracy. Since the decay of the effec-tive hopping matrix element with the impurity-impuritydistance is the slowest in 1d, we used a 1d host to maxi- mize the magnetic frustration induced by hopping matrixRe ∆ . Note that one of the main results, namely the ex-istence QCPs in the phase diagram, which emerge dueto FM correlations between the local moments, is not re-stricted to 1d systems. These QCPs originate from thefact, that we are studying MIAMs of the second kindwith rank( Γ ) ≤ < N f , which likewise exist in higherspatial dimensions as discussed in Sec. II D 1. We alsouse a simple nearest-neighbor tight binding model witha dispersion ǫ k = − t cos( ka ) + ǫ c . In case of an oddnumber N f we prevent a spin singlet formation alreadywithin the multi-impurity cluster.The N f impurities form a short finite size chain in1d. The orbital energies and the Coulomb repulsion U are chosen such that the local orbitals are approximatelysingly occupied at low temperature, and the induced localmoment is subject to the remaining interactions.By shifting the band center t ii = ǫ c from ǫ c = 0 to ǫ c → D = 2 t , we reduce the electron filling in the conductionband of the host, reduce the Fermi wave vector k f , andhence altering the matrix elements ∆ lm ( − i + ) such thatthe spin-spin correlation functions are associated with alonger wave length. While ǫ c = 0 refers to a half-filledparticle-hole symmetric conduction band, we approachthe one conduction electron limit for | ǫ c | /D → Γ ) = 2, with the exception of | ǫ c | = D where therank changes to one.Three points are worth to note. Such problems (i)can be treated by a two-band NRG approach as a conse-quence of rank( Γ ) ≤ S imp , is given by S imp = k B ln( N f )in the limit | ǫ c | → D in accordance with Ref. [44]. For N f >
2, we are (iii) always investigating a MIAM prob-lem of the second kind depicted as regime III in Fig. 1.In this section we start with dense impurity arrayswhere the correlated orbitals are placed next to eachother, such that the antiferromagnetic RKKY interactiondominates for a roughly half-filled conduction band. Westudy the effect of frustrated RKKY interaction at inter-mediate band-fillings and QCPs due to FM correlationsoccur near the band-edges for PH symmetric impuritiesand at intermediate fillings for large PH asymmetry. Fol-lowing this, we present NRG results for dilute impurityconfigurations with dominating ferromagnetic RKKY in-teractions for a half-filled conduction band, which ismore suited for ferromagnetic HF materials. These di-lute multi-impurity models exhibit FM correlations andthe associated QCPs in a realistic region of the param-eter space which might be connected to those found inthe quasi 1d ferromagnetic HF materials [20, 28, 29].Unless otherwise stated, we set the NRG discretizationparameter λ = 3 and kept N S = 5000 states after eachiteration.5
1. Kosterlitz-Thouless type Quantum Phase transitions andthe Phase diagram in dense impurity arrays
FIG. 6. (a) Impurity contribution to the entropy for T → U/ Γ = 10 , ǫ fi = − U/ D/ Γ = 10 . (b) The two spin-spin correlation functions for the same pa-rameter. The center impurity is labeled i = 2, the two outercorrelated sites i = 1 , Above, we introduced the parametrization of the theimpurity contribution to the entropy, S imp , in terms of anground state degeneracy g : S imp = k B ln( g ). This effec-tive degeneracy g = exp( S imp /k B ) is plotted as functionof | ǫ c | in Fig. 6(a) for N f = 3 and a particle-hole sym-metric Hamiltonian H corr . It shows the typical behaviorof the MIAM under investigation here. Starting froma singlet ground state, g = 1, the degeneracy increasesin integer steps to the maximum g = N f , which is al-ways reached for | ǫ c | /D →
1. The point of increase de-fines a quantum critical point (QCP) which is Kosterlitz-Thouless (KT) type as shown below.In Fig. 6, we concentrate to parameters close to thechanges of entropy. In order to understand the physicsof the two different quantum phases in the depicted re-gion of ǫ c , the two different spin-correlation functions areplotted in Fig. 6(b).In region I, S imp ( T →
0) vanishes, and the correlationfunction of the two outer impurity spins are anti-parallelwhile the central spin moment is screened by the Kondoeffect. This setup is equivalent to the triangular spincluster investigated in Ref. [92]. The correlation functionbetween the two outer spins with the central spin almostvanishes. The spin singlet formation involves two com-ponents: the AF interaction mediated by the dominating next-nearest hopping between two outer spins, and theKondo effect in the even sector that removes the localmoment of the central spin.The QCP occurs at the red point added to the hor-izontal axis in Fig. 6(a): At this point S imp jumps to k B ln(2) indicating a doubly degenerate ground state inregion II. The spin-spin correlation function, however,shows a smooth crossover at this point indicating thephysics of the new ground state. The nearest neighborspin-spin correlation function depicted in red in Fig. 6(b)rises from an anti-alignment to a finite ferromagnetic cor-relation. This observation is consistent with the notionthat a reduction of k f eventually changes the nearestneighbor RKKY interaction from AF to FM. The next-nearest neighbor spin-spin correlation function changessign to ferromagnetic correlations of the same magni-tude: All three local moments of the correlated clusterare ferromagnetically coupled in this parameter regimeand form a large I = 3 / Γ ) = 2,we are in an underscreened Kondo regime. We found atwo stage Kondo effect quenching the I = 3 / I = 1 / Γ ) = 1 atthe point | ǫ c | /D = 1: the total I = 3 / S imp /k B = ln(3).The black entropy curve in Fig. 6(a) suggests that thisis a smooth transition at finite temperature, governedby a crossover value | ǫ c | /D <
1. This data, however,was obtained for a fixed number of NRG iteration thatcorrespond to a fixed temperature of
T / Γ = 10 − . Weincreased the number of NRG iteration and added the re-sults as light-blue curve representing a temperature thatis 20 orders of magnitude lower, i. e. T / Γ = 10 − . Thecrossover region is clearly pushed closer to | ǫ c | /D = 1.The finding in region II of Fig. 6(a) can be understoodin terms of a band filling dependent low-energy scale T L ( ǫ c ) that governs the crossover from the unstable localmoment fixed point with I = 3 / S imp = k B ln(2).This low-energy scale can be associated with the Kondo-screening due to the smaller coupling Γ n that vanishesfor | ǫ c | → D . To access the entropy of the low-energyfixed point requires that T ≪ T L ( ǫ c ). From our NRGdata we can conclude thatlim | ǫ c |→ D T L ( ǫ c ) = 0 (30)and, therefore, the critical point (i) is located at | ǫ c | /D =1 and (ii) is of Kosterlitz-Thouless type since it vanishesas T L ( ǫ c ) ∝ exp( A/ p D − | ǫ c | ) with some fitting param-eter A .6 FIG. 7. S imp as function of | ǫ c | /D for (a) N f = 4 and (b) N f = 5 and a fixed temperature T / Γ = 10 − (correspond-ing to a fixed number of NRG iterations N = 50.) for particle-hole symmetric impurities with U/ Γ = 10. For N f = 4 wefind three regions and three KT quantum phase transitionpoints, for N f = 5 four regions separated by a KT type phasetransition are identified. The investigation of the low-energy fixed points of theMIAM with N f = 4 and N f = 5 reveals a similarpicture as shown in Fig. 7: for a half-filled conductionband we always find a vanishing S imp for the stable low-temperature fixed point that can be interpreted as a spin-singlet ground state formation. In the limit | ǫ c | → D , wereproduce the prediction of Sigrist et al. [44, 93] even inour finite size system: an unstable local moment fixedpoint with I = N f / I = ( N f − /
2, corresponding to S imp = k b ln( N f ), since rank( Γ ) = 1 at this point inde-pendent of N f . The ground state degeneracy g rises ininteger steps from g = 1 to g = N f , and the correspond-ing QPTs are of KT type.In each of the regions one additional moment is alignedferromagnetically with the others: The self-screening ofthe local impurity spins via the RKKY mediated inter-action becomes less and less effective since this interac-tion becomes predominating ferromagnetic for low bandfillings. The hopping matrix elements Re∆ lm ( ǫ c ) reachtheir maximum at | ǫ c | = D due to the maximal particle-hole asymmetry of the conduction band. At the sametime, the density of states diverges at the band edge in1d, such that the hybridization to the conduction bandsstart to dominate resulting in a FM alignment of the .
05 0 . .
15 0 . T / C ( ε cc − ε c ) / D ε f / Γ = − ε f / Γ = − ε f / Γ = − ε f / Γ = − − . − . − . − . − . − − . − − . − − . − ε cc / D ε f / Γ FIG. 8. (a) Low temperature scale T for particle-hole asym-metric three impurity models with U/ Γ = 50 , D/ Γ = 10and four different values of ǫ f . (b) the critical value ǫ cc asfunction of ǫ f . spins [31, 32].Depleting the conduction band more and more, leavesasymptotically a Hubbard type model [72] where close tohalf-filling the FM aligned local moments could be inter-preted as a precursor of the Nagaoka mechanism [93] toferromagnetism in the Hubbard model. Energy is gainedby allowing the impurity electrons to delocalized in a uni-form polarized background.We still need to prove the claim that the transitionsbetween the different regions in Fig 6(a) and Fig. 7, indi-cated by the red dots on the horizontal axis, are indeedQCPs of the KT type. We exemplify this point by inves-tigating the low temperature scale T ( ǫ c ) as a functionof ǫ c in the three impurity problem, N f = 3, for fourdifferent values of ǫ f and a fixed U/ Γ = 50. We selectedthese parameters to demonstrate that the transition typeis unrelated to the particle-hole symmetry and that thecritical value of ǫ cc depends on the particle-hole asymme-try of the correlated cluster. In Ce, for instance, the 4f-shell occupation fluctuates between zero and one, so thatthe limit U → ∞ was successfully employed [67, 94, 95]to understand the basic properties of such systems.The low-temperature scale T ( ǫ c ) is defined via thecrossover from the last unstable local moment fixed pointto the stable low-temperature fixed point when S imp ( T )7reaches the mid point between both entropies: S imp ( T ) = 12 ( S LMimp + S LTimp (0)) . (31)We fitted the T ( ǫ c ) to the exponential form T ( ǫ c ) = C e C √ ǫcc − ǫc ) (32)close to the transition point and extracted the transitionpoint ǫ cc as well as the parameters C and C . We plottedthe NRG data T ( ǫ c ) /C as a function of ( ǫ cc − ǫ c ) /D inFig. 8(a). In addition we added the fitting curve definedin Eq. (32) as black solid lines for the four cases demon-strating an excellent agreement with the NRG data withthis analytical form. This can be done in the vicinityof all QCPs for different cluster sizes N f . Therefore, allQCPs at the critical values indicated by red or black dotsin Figs 6(a) and Fig. 7 are of the KT universality class.We added the particle-hole asymmetry dependency of ǫ cc as Fig. 8(b). Upon increasing ǫ f from ǫ f / Γ = − ǫ f / Γ = − ǫ cc is significantly reduced.
2. The role of magnetic frustration in the S imp = 0 phase After establishing a step wise increase of the fixed pointdegeneracy g in the limit | ǫ c | /D → ǫ c parameter space in this section. Thisregime is determined by S imp = 0, indicating a singletground state of the low-energy fixed point.The notion of a competition between an AF RKKYscreening and the Kondo screening by the two conductionchannels was establish in the two-impurity model [31]. Inthis MIAM of first kind, there are always enough conduc-tion electron channels available to Kondo screen all thelocal moments. For N f > Γ ) < N f , the topol-ogy of the model becomes different and such a scenariois not applicable any more. The concept of competingKondo effect and RKKY interaction, both generated bythe host conduction electrons, become less meaningfuldue to the lack of available screening channels. For peri-odic systems, Eq. (6) suggests that the rank( Γ ) is of theorder of the ~k -points on the Fermi surface and, therefore,is much smaller then N f .We begin with the three impurity model and calcu-late the two low-energy crossover temperatures to the S imp = 0 stable fixed point: T ( ǫ c ) denotes the crossovertemperature to the last unstable S = 1 / T ( ǫ ) characterizes the approach to the stable low-energy fixed point. The temperature T replaces T K inthe single-impurity model since it is associated to theKondo effect. The results are plotted in Fig. 9(a). Wenote that both crossover temperatures change continu-ously with ǫ c but develop a cusp at | ǫ c | /D = 0 . T remains finite over the whole parameter regime until theend at about | ǫ c | /D ≈ .
945 where the KT type QPT toa stable fixed point with S imp /k B = ln(2) occurs that was . . . . N f = T / Γ | ε c | / D T / Γ T / Γ ∆ ( z ) ≈ ℜ∆ ( ε c ) − . − . − . − . . . . . . h ~ S ~ S i i | ε c | / D h ~ S ~ S ih ~ S ~ S i FIG. 9. (a) The two low temperature scales T and T vs ǫ c in the three impurity model. T (blue) is defined as crossovertemperature approaching the unstable LM fixed point with I = 1 / T (red) denotes the crossover temperature fromthe unstable S = 1 / D ~S ~S E and D ~S ~S E vs ǫ c . Parameters: particle-hole symmetric impurities with U/ Γ = 30 , ǫ fl = − U/ D/ Γ = 10. discussed in the previous section. The overall decrease in T ( ǫ c ) is related to the dependency of the effective changeof fluctuation strengths Γ n as function of ǫ c : the couplingto the relevant orbital involved in the Kondo screeningdecreases with increasing | ǫ c | .The spin-spin correlation functions calculated at thelow-energy fixed point reveals that two different regionsemerge as shown in Fig. 9(b). For small values | ǫ c | /D , thenearest neighbor spins are AF aligned while the two outerimpurity spin are FM correlated. The spin-spin correla-tion functions indicate that the three local spins alwaysadd up to a S = 1 / FIG. 10. Schematic diagram of the tight binding hoppingparameters between the tree impurities occurs on the temperature scale T . One of the two con-duction electron channels is sufficient to Kondo screenthe remaining S = 1 / T wherethe impurity entropy is removed. For 0 . < | ǫ c | /D , thetwo local spins connected to the same sublattice are AFaligned while the center spin is screened by the Kondoeffect.In order to shed some light into the nature of the ob-served cusp, | ǫ c | /D ≈ .
5, we solved the three impuritycluster model for Γ = 0, i. e. by decoupling the impuritiesfrom the two effective conduction bands, using exact di-agonalization. We calculated the crossover temperatureto the low-energy ground state from the temperature de-pendence of the entropy S imp ( T ). This crossover scalewas added as continuous black line to Fig. 9(a) and tracesthe NRG temperature T very well. At the cusp, how-ever, this crossover temperature vanishes in the decou-ple cluster model: a level crossing between two differenttwofold degenerate cluster ground states occurs whichcan be characterized by different spin-spin correlationsfunction. This defined a point of maximal magnetic frus-tration in the system.The nature of those local ground states can be un-derstood in terms of the effective hopping parametersin the mapped model as shown schematically in Fig 10.By only considering Re∆ ( ǫ c ), we obtain from the or-bitals 1 and 3, two molecular orbitals that have evenor odd symmetry under a parity transformation can beconstructed. The even orbital couples to the orbital 2via Re∆ ( ǫ c ) and Re∆ ( ǫ c ) and two new even orbitalsare generated. Then we fill these orbitals with three elec-trons. For | ǫ c | /D < .
5, the one of the even orbitals hasthe lowest energy and is filled with two electrons forminga singlet. The third electron occupies the orbital withodd symmetry. Therefore, the local moment is locatedat the outer edge on the orbitals 1 and 3 in real space.For | ǫ c | /D > .
5, Re∆ ( ǫ c ) dominates and the odd sym-metry orbital has the lowest single particle energy, andis doubly occupied. This singlet formation results in astrong AF correlation function D ~S ~S E in this regime.The local moment is located in the higher lying even or-bital, and therefore, more localized in the orbital 2.At the degeneracy point, all three hopping matrix ele-ments in Fig 10 become equal and generate a maximallymagnetically frustrated system. A level crossing between these two doublet ground states occurs forming a four-fold degenerate cluster ground state. This emerging localpicture also explains the observed change in the spin-spincorrelation function shown in Fig. 9(b) for the full prob-lem. For | ǫ d | /D > .
5, the nearest neighbor spin cor-relations are suppressed and the central spin is Kondoscreened.Although the real-space geometry of our cluster isa short three-site chain, the hopping matrix elementsRe∆ ij generate a trimer with degenerate AF Heisenbergcouplings as investigated in the literature [33–35, 92].The different physics found here is related to the reducednumber of screening channels: While the Refs. [33–35] in-vestigate a Kondo trimer model of first kind, we derivedan example of a trimer model of second kind [92].Right at the degeneracy point, both doublets with dif-ferent parity symmetry couple to one of the two effec-tive conduction bands. If we assume PH symmetry andidentical couplings Γ n , this would result in the isospinKondo regime found in Refs. [33, 34] and already dis-cussed in Sec. III A. However, the effective tunneling ele-ments Re∆ ij , which generate the AF RKKY interactionand, therefore, are necessary in order to obtain magneticfrustration, break the PH symmetry and cause a FL withspin-singlet ground state, since this is a relevant pertur-bation of the isospin Kondo FP [33, 34].Furthermore, the mapped MIAM lacks the helicitysymmetry of the C group in general, and, consequently,the two couplings Γ n of the degenerate doublets are notidentical. In this case, the cluster ground state degen-eracy is lifted at the degeneracy point by the asymmet-ric coupling to the two conduction band channels, suchthat even the coupling of only one of the conduction elec-tron channels to one of the two degenerate doublets canquench the remaining cluster entropy. The energy split-ting of both doublets, however, is dynamically generatedby the RG procedure and remains small. Therefore, wedo not observe a two-stage screening process: only onelow temperature scale T emerges, even at the degener-acy point, defining the crossover from the LM fixed pointwith S imp ≈ k B ln(4) to the stable strong coupling fixedpoint with S imp = 0.At the degeneracy point, the definition of T becomesobsolete since we observe a direct crossover from a four-fold degenerate unstable local moment fixed point to thesinglet ground state. Therefore, we do not find anotherQCP in the S imp = 0 regime.While the low-energy scale T decreases with increas-ing | ǫ c | /D due to the change in the eigenvalues Γ n thatcouple to the impurity ground state multiplet, the en-hancement of the magnetic fluctuations in the vicinityof the degeneracy point causes the cusp in T signalingthe adiabatic change of the spin correlations in the low-energy fixed point. The crossover temperature T ( ǫ c ) andthe location of the cusp is symmetric in ± ǫ c , for particle-hole symmetric impurities, qualitatively the same phasesare observed for particle-hole asymmetric impurities butwith asymmetric curves, not shown here.9 . . . . N f = . . . . T / Γ | ε c | / DT / Γ T / Γ T / Γ | ε c | / D ∆ ( z ) ≈ ℜ∆ ( ε c ) FIG. 11. The two low temperature scales T and T vs ǫ c in the N f = 5 MIAM. T (blue) denotes crossover temper-ature approaching the lowest unstable LM fixed point and T (red) crossover temperature from the unstable LM fixedpoint to the singlet fixed point. The crossover temperaturefor the multi-impurity system without coupling to the con-duction band channels is added as black line. Parameters: asin Fig. 9 A similar picture emerges for N f = 5 in the S imp = 0regime, as shown in Fig. 11. For even N f , the impuritycluster can form a singlet ground state without couplingto the conduction bands: Hence we focus on odd N f where the interplay between the RKKY interaction andthe Kondo effect is relevant. Clearly visible are the con-tinuous change of T ( ǫ c ) but now showing four cusps.Since the total spin is conserved in the model, the im-purity cluster eigenstates can be classified by the totalangular momentum. The hopping matrix Re∆ ij ( ǫ c ) in-duces AF interactions between the spins, so that the clus-ter ground state is located in the subspace with the lowestangular moment. Adding N f = 5 local moments S = 1 / J = 1 / ǫ c is chang-ing the energy spectrum of the five J = 1 / J = 1 /
2. The black solid line for T added to Fig. 11 was calculated from the impuritycluster spectrum including the Coulomb interaction butneglecting the coupling to conduction electrons channelsin Eq. (11).The cluster calculation demonstrates that the cuspsobtained from the full model are associated with a changeof the cluster ground state in the unstable intermediatefixed point. Note, however, that in the impurity clusterthe scale T vanishes at the four level crossing points ofthe ground states, which is not visibly here since we didnot zoom into the degeneracy point with high resolution.For the five different J = 1 / S imp = k B ln(4) is characterized bythe crossover temperature T . Quantum fluctuations en-hance T at the degeneracy point and we also found, thata single spin degenerate band is sufficient to quench thelocal moments even at the degeneracy point of the impu-rity cluster, where the local system is magnetically frus-trated.
3. Kosterlitz-Thouless type Quantum Phase transitions andthe Phase diagram in dilute impurity arrays
In dense impurity arrays, where the correlated orbitalsare located next to each other, the antiferromagneticRKKY interaction typically dominates the interactionbetween the local moments of the correlated orbitals fora roughly half-filled host conduction band. In order toqualitatively simulate ferromagnetic HF materials [17–24, 28–30], we study the dilute impurity configurationalready discussed in Sec II F 2 and schematically depictedin Fig. 3. In this case, the correlated orbitals only hy-bridize with the host Wannier orbitals of one of two sub-lattices in a bipartite lattice. In Sec. II F 2 we ascribedthe ferromagnetic ground state at half-filling, ǫ c = 0,obtained by DMRG calculations [42, 43, 89, 90], to theabsence of an effective tunneling between the correlatedorbitals Re ∆ ( ǫ c = 0) = 0 and rank( Γ ) < N f .In the non-interacting limit ( U = 0), the d-dimensional, depleted PAM can be exactly diagonalized,resulting in three bands. At particle-hole symmetry, cor-responding to half-filling, one of these bands is totallyflat, leading to a high degeneracy of the ground state,and posses mainly f -character for small couplings V /D .As demonstrated by a first-order perturbation theory in U , weak interactions within this flat band result in a fullypolarized ground state of the model [89]. For 1d and2d, it was further shown, that this polarized state per-sists to arbitrary strengths of the Coulomb interaction U [42, 43, 89, 90].From the perspective of our low-energy mapping, thisresult is just a consequence of the absence of the de-localizing, band generated effective hopping matrix el-ements Re∆ ij ( ǫ c = 0) = 0 in combination with anextreme reduction of available conduction band screen-ing channels due to rank( Γ ) = 1. However, for any ǫ c = 0, the flat band becomes dispersive, tantamountwith the appearance of delocalizing tunneling elementsRe∆ ij ( ǫ c = 0) = 0 in the mapped Hamiltonian. Thisdelocalizing terms might lead to a localized/delocalizedMott-Hubbard insulator transition in the strongly inter-acting limit of the depleted PAM, with a possible con-nection to the emerging NFL behavior at the FM QCPin heavy Fermion ferromagnets [20, 28, 29].In order to qualitative study this competition betweenthe AF and FM RKKY interaction, we analyze the de-pleted Anderson model for a finite number of N f = 30 FIG. 12. exp( S imp /k B ) phase diagram for (a) N f = 3 and (b) N f = 4 impurities, separated by ∆ R = 2 a and plotted against ǫ c /D and U/ Γ for a constant temperature T / Γ = 10 − , ǫ fl / Γ = − D/ Γ = 10. and N f = 4 correlated orbitals in 1d, the simplest MI-AMs of the second kind. However, as already discussed,such QCPs generally occur in MIAMs of the second kindand, therefore, are not restricted to 1d.In Fig. 12, we color plotted the degeneracy g =exp( S imp /k B ) for the dilute MIAM at a fixed temperature T / Γ = 10 − with (a) N f = 3 and (b) N f = 4 impuri-ties, as a function of the band-filling ǫ c and the strengthof the Coulomb interaction U , with ǫ f / Γ = − D/ Γ = 10. The phase diagram for N f = 3 and N f = 4correlated orbitals is qualitatively identical, separating aspin compensated singlet phase at finite ǫ c (black) froma phase with finite degeneracy around ǫ c = 0 (yellow redblue).The maximal value of the degeneracy near ǫ c = 0 (yel-low) is exp( S imp /k B ) = 3 for N f = 3 in Fig. 12(a),and exp( S imp /k B ) = 4 for N f = 4 in Fig. 12(b). At T = 0, however, this value color coded in yellow canonly be reached at precisely ǫ c = 0, where rank( Γ ) = 1:for a very small but finite | ǫ c | >
0, rank( Γ ) = 2 suchthat g = g max −
1. Since one of the two effective hy-bridizations, Γ n , is very small for small deviations fromhalf-filling, the associated Kondo temperature is expo-nentially suppressed. Consequently, the extended yellowregion of the maximal degeneracy is only replaced bya region with g max −
1, once the temperature is lowerthan the exponentially suppressed lowest Kondo tem-perature. However, even at T = 0, two/three regionswith different degeneracy of the ground state remain for N f = 3 /N f = 4.As discussed in Sec. III B 1, regions with different de-generacy of the ground state are separated by a QCP ofKosterlitz-Thouless type and associated with the decou-pling of an effective spin-1 / ǫ c for PH symmetric impurities ( U = − ǫ f = 6Γ inFig. 12). In case of U < − ǫ f , the degenerate phaseis shifted into the ǫ c /D < U > − ǫ f .In the two impurity model, a spin-singlet ground stateis always generated for finite ǫ c since enough screeningchannels are available. The emergence of QCPs due toFM correlations, as reported in this and the previoussection, however, are a common feature of MIAMs of thesecond kind. Since the phase diagram for N f = 3 and N f = 4, shown in Fig. 12, is qualitatively identical for N f = 5 (not shown here), and the physical origin of theQCPs is just a consequence of a competition betweenAF and FM RKKY interactions in combination with thereduced number of available conduction band screeningchannels, the phase diagram will be qualitatively identi-cal if more correlated orbitals are added. The geometricarrangement of the correlated orbitals onto the latticegoverns the hybridization matrix ∆ ( z ). While the FMQCPs occur in depleted host bands, | ǫ c /D | →
1, in denseimpurity setups, they are shifted to half-filling in diluteimpurity arrays. First-order perturbation theory in U re-vealed that the FM ground state in the depleted model athalf-filling also exists in higher spatial dimensions of thelattice [89], and is not an artifact of a 1d host geometry.Altogether, these results indicate, that a transition froma delocalized paramagnetic to a localized ferromagneticstate, which is beyond a HertzMillisMoriya spin densitywave scenario, might be realizable in the depleted PAMwith ǫ c = 0. Due to the exponential suppression of thelowest crossover scale in such models, the unstable in-termediate fixed points might be more relevant for theexperimentally accessible temperatures. IV. CONCLUSION AND OUTLOOK
We presented a mapping of a strongly correlated multi-impurity models onto a correlated cluster model subjectto couplings to a number of effective conduction bands.The AF exchange interaction is encoded in the clusterorbital hopping matrix, whereas the FM interaction andthe competing Kondo effect by the remaining host de-grees of freedom are included in the coupling to the effec-tive conduction band channels. This allows us to studythe self-screening of the local moments as well as theemerging local moment fixed points via exact diagonal-ization of the cluster prior to a investigation of the fullproblem. It also opens the door for determining the re-1quirements in the models with potential magnetic frus-trations such as the trimer model.Since our mapping incorporates the RKKY interactionand the related potential scattering on an equal footing,we were able to show that the frustration induced NFLfixed point of a trimer model with C symmetry [33, 34] isrestricted to only a finite range of AF RKKY interactionstrength, and can completely disappear in the phase di-agram if the Kondo temperature exceeds a certain limit.For large N f , the MIAMs fall typically into problems ofthe second kind, where the number of impurities exceedsthe number of available Kondo screening channels. Themost prominent example is the PAM. Since the maximalnumber of independent conduction bands that couple tothe correlated cluster scales with the number of k -pointson the Fermi surface, it is limited to N b = 2 in 1d. As aconsequence, our mapped model predicts a local momentfixed point for the dilute Anderson lattice in 1d [42, 43]with a large local moment of I = ( N f − / / T at the point of magnetic frustration could be misread as a indi-cation of a QCP between different ground states. NRGcalculations reveal, however, that there is still a smoothcrossover between two different singlet states since theasymmetry is a relevant perturbation for the frustratedisospin Kondo regime [33, 34], just as been shown for thetwo-impurity problem [11, 12]. At the band edges theferromagnetic correlations are recovered even for a smallfinite size impurity cluster, and the system shows a se-ries of QCPs of KT-type when changing the ground statedegeneracy by one.There is an ongoing debate about a confine-ment/deconfinement transition in HFs where magneticorder might reduce the number of conduction electronscontributing to the Fermi surface: a heavy FL should becharacterized by a large Fermi surface while in a local-moment magnetic metal the correlated electrons are ex-cluded from the Fermi volume [4]. Such a scenario of frac-tionalized Fermi liquids [96] requires a two fluid model,where the well defined light quasiparticles and a spin-liquid are disconnected. This might be achievable by aMott-Hubbard insulator transition within the correlatedelectron subsystem, in which the remaining spin-spin in-teractions induce a spin liquid. The lack of charge fluc-tuation channels suppresses the coherent quasiparticleformation decoupling the light quasiparticles on a low-energy scale.Recent experiments [28–30] revealed strange-metal be-havior at the FM-PM transition in HFs, and from thetheoretical point of view it is known that the FM Kondolattice posses a small Fermi surface excluding the localmoments [97]. Motivated by theses results, we investi-gated dilute multi-impurity models of the second kind,where the FM RKKY interaction dominates for a roughlyhalf filled conduction band, and demonstrated the exis-tence of several QCPs of KT-type at which local momentsdecouple from the continuum. This finding indicates apossible confinement/deconfinement transition in a peri-odic extension of such models, the depleted PAM.Another fascinating subject is the physics of Kondoholes [98–101]. 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