Kondo Destruction in RKKY-Coupled Kondo Lattice and Multi-Impurity Systems
aa r X i v : . [ c ond - m a t . s t r- e l ] M a r Kondo Destruction in RKKY-Coupled Kondo Lattice and Multi-Impurity Systems
Ammar Nejati, Katinka Ballmann, and Johann Kroha
1, 2, ∗ Physikalisches Institut and Bethe Center for Theoretical Physics,Universität Bonn, Nussallee 12, D-53115 Bonn, Germany Center for Correlated Matter, Zhejiang University, Hangzhou, Zhejiang 310058, China (Received December 8, 2016; published March 8, 2017)In a Kondo lattice, the spin exchange coupling between a local spin and the conduction electronsacquires nonlocal contributions due to conduction electron scattering from surrounding local spinsand the subsequent RKKY interaction. It leads to a hitherto unrecognized interference of Kondoscreening and the RKKY interaction beyond the Doniach scenario. We develop a renormalizationgroup theory for the RKKY-modified Kondo vertex. The Kondo temperature, T K ( y ) , is suppressedin a universal way, controlled by the dimensionless RKKY coupling parameter y . Complete spinscreening ceases to exist beyond a critical RKKY strength y c even in the absence of magneticordering. At this breakdown point, T K ( y ) remains nonzero and is not defined for larger RKKYcouplings, y > y c . The results are in quantitative agreement with STM spectroscopy experimentson tunable two-impurity Kondo systems. The possible implications for quantum critical scenariosin heavy-fermion systems are discussed. The concept of fermionic quasiparticles existing even instrongly interacting many-body systems is fundamentalfor a wealth of phenomena summarized under the termFermi liquid physics. In heavy-fermion systems [1], quasi-particles with a large effective mass are formed by theKondo effect [2]. The conditions under which these heavyquasiparticles disintegrate near a quantum phase transi-tion (QPT) have been an important, intensively debatedand still open issue for many years [1].The heavy Fermi liquid, like any other Fermi liquid,may undergo a spin density-wave (SDW) instability, lead-ing to critical fluctuations of the magnetic order param-eter but leaving the heavy quasiparticles intact. Thisscenario is well described by the pioneering works ofHertz, Moriya and Millis [3–5]. However, early on Do-niach pointed out [6] that the Kondo spin screening ofthe local moments should eventually cease and give wayto magnetic order, when the RKKY coupling energy be-tween the local moments [7–9] becomes larger than thecharacteristic energy scale for Kondo singlet formation,the Kondo temperature T K . It is generally believed thatthe Kondo destruction is driven by the critical fluctua-tions near a QPT. Several mechanisms have been pro-posed, invoking different types of fluctuations, includingcritical fluctuations of the local magnetization couplingto the fermionic quasiparticles (local quantum criticality)[10, 11] and Fermi surface fluctuations self-consistentlygenerated by the Kondo destruction [12]. Most recently,a scenario of critical quasiparticles with diverging effec-tive mass and a singular interaction, induced by criti-cal antiferromagnetic fluctuations, has been put forward[13–15]. Intriguing in its generality, it does, however, notinvoke Kondo physics.Here, we show that the heavy-electron quasiparticlescan be destroyed by the RKKY interaction even without ∗ Email: [email protected] critical fluctuations. This occurs because of a hithertounrecognized feedback effect: in a Kondo lattice or multi-impurity system, the RKKY interaction, parametrizedby a dimensionless coupling y , reduces the Kondo screen-ing energy scale T K ( y ) . This reduction implies an in-crease of the local spin susceptibility at low temperatures T , χ f ( T = 0) ∼ /T K ( y ) , which in turn increases theeffective RKKY coupling. We derive this effect and an-alyze it by a renormalization group (RG) treatment. Inparticular, we calculate the temperature scale for Kondosinglet formation in a Kondo lattice, T K ( y ) . It is sup-pressed with increasing y in a universal way. Beyonda critical RKKY coupling y c , complete Kondo singletformation ceases to exist. However, at this breakdownpoint T K ( y c ) remains finite, and the suppression with re-spect to the single-impurity Kondo scale takes a universalvalue, T K ( y c ) /T K (0) = 1 /e , where e = 2 . ... is Euler’sconstant. These findings are consistent with conformalfield theory results [16, 17] and in quantitative agreementwith STM spectroscopy experiments on tunable, RKKY-coupled two-impurity Kondo systems [18, 19].The present results directly apply to cases where long-range order does not play a role, that is, two-impurityKondo systems [18–20], compounds where the magneticordering does not occur at the Kondo breakdown point[21], and temperatures sufficiently above the Néel tem-perature [22]. They will set the stage for a completetheory of heavy-fermion quantum criticality by includingcritical order-parameter fluctuations either of the incom-pletely screened magnetic moments or of an impendingSDW instability. The model. – We consider the Kondo lattice model H = X k ,σ ε k c † k σ c k σ + J X i S i · s i (1)where c k σ , c † k σ denote the conduction ( c ) electron oper-ators with dispersion ε k . S i are the local spin operatorsat the lattice sites x i , exchange coupled to the conduc-tion electron spins s i = P σ,σ ′ c † iσ σ σσ ′ c iσ ′ via an on-site, γ RKKY(d) χ f jji RKKY(x) γ f χ jj ii J = ji (a) ++ Ω−Ω i ij’j
Ω−Ω (b) + ij ij’ Ω Ωω−Ω ω ωω+Ωωω Γ cf Γ Γ cf cf
Γ Γ cfcf
FIG. 1: (a) f -spin– c -electron vertex ˆΓ cf , composed of the on-site vertex J at site i and the RKKY-induced contributionsfrom surrounding sites j = i to leading order in the RKKYcoupling: γ ( d )RKKY (direct term) and γ ( x )RKKY (exchange term).(b) One-loop diagrams for the perturbative RG. Solid lines:electron Green’s functions G c . Dashed lines: pseudofermionpropagators G f of the local f spins. The red bubbles representthe full f -spin susceptibility at sites j . antiferromagnetic coupling J > . The local spins willhenceforth be termed f spins, as they are typically real-ized in heavy fermion systems by the rare-earth f elec-trons. We will use the pseudofermion representation ofthe f spins, S i = 1 / P τ,τ ′ f † iτ σ ττ ′ f iτ ′ , with σ the vec-tor of Pauli matrices and f iσ , f † iσ ′ fermionic operatorsobeying the constraint ˆ Q = P σ f † iσ f iσ = 1 . It is cru-cial that the coupling between different f spins is nota direct exchange interaction, but mediated by the con-duction band [7–9] and generated in second order by thesame spin coupling J that also creates the Kondo ef-fect. The essential difference can be seen from the ex-ample of a two-impurity Kondo system, S , S : with adirect impurity-impurity coupling K S · S , and for a spe-cific particle-hole symmetry [16], this model can exhibita dimer singlet phase where the dimer is decoupled fromthe conduction electrons (scattering phase shift at theFermi energy φ dimer = 0 ). As a function of K , this dimersinglet phase is then separated from the Kondo singletphase (scattering phase shift φ Kondo = π/ ) by a quan-tum critical point (QCP) [16, 23], see also Ref. [17]. Bycontrast, when the interimpurity coupling is controlled bythe RKKY interaction only, i.e. generated by J , a de-coupled dimer singlet and, hence, a second-order QCP isnot possible. Instead, we find below that the Kondo sin-glet formation at T = 0 breaks down at a critical strengthof the RKKY coupling, however without a diverging lo-cal impurity susceptibility, that is, with a discontinuousjump of T K ( y ) . The profound implications of this behav-ior will be discussed below. RKKY-coupled c-f vertex and renormalization group .–We develop an analytical renormalization group for RKKY-coupled Kondo multi-impurity and lattice sys-tems, taking the proper renormalizations of all appearingvertices into account. The RKKY vertex ˆΓ ff couplingtwo f spins has no logarithmic RG flow, since the re-coil (momentum integration) of the itinerant conductionelectrons prevents an infrared singularity of the RKKYinteraction. ˆΓ ff thus remains in the weak couplingregime. The formation of the strong-coupling Kondo sin-glet, which is the origin of the heavy-Fermion state, issignalled by a RG divergence of the spin-scattering vertexoperator ˆΓ cf between c electrons and an f spin. In thecase of multiple Kondo sites, this vertex acquires nonlocalcontributions in addition to the local coupling J at a site i , because a c electron can scatter from a distant Kondosite j = i , and the spin flip at that site is transferred tothe f spin at site i via the RKKY interaction. In thisway, ˆΓ ff will influence the RG flow of ˆΓ cf , even though itis not renormalized itself. The corresponding diagramsare shown in Fig. 1 (a). As seen from the figure, sucha nonlocal scattering process necessarily involves the ex-act, local dynamical f -spin susceptibility χ f ( i Ω) on site j . The resulting c − f vertex ˆΓ cf has the structure ofa nonlocal Heisenberg coupling in spin space. The ex-change diagram, γ ( x )RKKY in Fig. 1 (a), contributes only asubleading logarithmic term as compared to γ ( d )RKKY [24].In particular, it does not alter the universal T K ( y ) sup-pression derived below and can, therefore, be neglected.To leading (linear) order in the RKKY coupling, ˆΓ cf thusreads (in Matsubara representation), ˆΓ cf = h Jδ ij + γ ( d ) RKKY ( r ij , i Ω) i S i · s j (2) = (cid:2) Jδ ij + 2 JJ (1 − δ ij ) χ c ( r ij , i Ω) ˜ χ f ( i Ω) (cid:3) S i · s j , where r ij = x i − x j is the distance vector between thesites i and j , and Ω is the energy transferred in the scat-tering process. χ c ( r ij , i Ω) is the c electron density cor-relation function between sites i and j [bubble of solidlines in Fig. 1 (a)] and ˜ χ f ( i Ω) := χ f ( i Ω) / ( g L µ B ) with g L the Landé factor and µ B the Bohr magneton. Notethat Eq. (2) contains the running coupling J at site i ,which will be renormalized under the RG, while at thesite j , where the c electron scatters, the bare coupling J appears, since all vertex renormalizations on that site arealready included in the exact susceptibility χ f . Higherorder terms, as for instance generated by the RG [seebelow, Fig. 1 (b)], lead to nonlocality of the incomingand outgoing coordinates of the scattering c electrons, x j , x j ′ , but the f -spin coordinate x i remains strictly lo-cal, since the pseudofermion propagator G f ( iν ) = 1 /iν is local [26]. For this reason, speaking of Kondo sin-glet formation on a single Kondo site is well defined evenin a Kondo lattice, and so is the local susceptibility χ f of a single f spin. The corresponding Kondo scale T K on a site j is observable, e.g., as the Kondo resonancewidth measured by STM spectroscopy on one Kondo ionof the Kondo lattice. The temperature dependence of thesingle-site f -spin susceptibility is known from the Bethe y / y c T K ( y ) / T K ( ) u T K ( y ) / T K ( ) Lo c a l m o m en t s FIG. 2: Universal dependence of T K ( y ) /T K (0) on the nor-malized RKKY parameter y/y c , solution of Eq. (9). The in-set visualizes the solution of Eq. (9) graphically. Black, solidcurve: right-hand side of Eq. (9). Blue line: left-hand sidefor y < y c . Red line: left-hand side for y = y c (where thered line and black curve touch). It proves that there is a crit-ical coupling y c beyond which Eq. (9) has no solution, and T K ( y c ) /T K (0) = 1 /e . ansatz solution [27] in terms of the Kondo scale T K . Ithas a T = 0 value χ f (0) ∝ /T K and crosses over tothe /T behavior of a free spin for T > T K . These fea-tures can be modeled in the retarded or advanced, local,dynamical f -spin susceptibility χ f (Ω ± i as χ f (Ω ± i
0) = ( g L µ B ) WπT K p /T K ) (cid:18) ± iπ arsinh Ω T K (cid:19) (3)where W is the Wilson ratio, and the imaginary part isimplied by the Kramers-Kronig relation.We now derive the one-loop RG equation for the c − f vertex ˆΓ cf , including RKKY-induced, nonlocal contribu-tions. The one-loop spin vertex function is shown dia-grammatically in Fig. 1 (b). Using Eq. (2), the sum ofthese two diagrams is up to linear order in the RKKYcoupling Y ( r ij , iω ) = (4) − J T X i Ω h Jδ ij + γ ( d ) RKKY ( r ij , i Ω) + γ ( d ) RKKY ( r ij , − i Ω) i × [ G c ( r ij , iω − i Ω) − G c ( r ij , iω + i Ω)] G f ( i Ω) . Here, ω is the energy of the incoming conduction elec-trons, G c ( r ij , iω + i Ω) is the single-particle c -electronpropagator from the incoming to the outgoing site.For example, for an isotropic system, G c ( r , ω ± i
0) = − πN ( ω ) e ± ik ( ε F + ω ) r /k ( ε F + ω ) r , with the bare densityof states N ( ω ) , and k ( ε F + ω ) the modulus of the mo-mentum corresponding to the energy ω .For the low-energy physics, the vertex renormalizationfor c electrons at the Fermi surface is required. This means setting the energy iω → ω = 0 + i and Fouriertransforming the total vertex Y ( r ij , iω ) with respect tothe incoming and outgoing c electron coordinates, x j , x i , and taking its Fourier component for momenta at theFermi surface k F , see Ref. [24]. Note that at the Fermienergy Y ( k F , is real, even though the RKKY-induced,dynamical vertex γ ( d )RKKY ( ± i Ω) appearing in Eq. (4) iscomplex valued [24]. This ensures the total vertex opera-tor of the renormalized Hamiltonian is Hermitian. By an-alytic continuation, the Matsubara summation in Eq. (4)becomes an integration over the intermediate c electronenergy from the lower and upper band cutoff D to theFermi energy ( Ω = 0 ). The coupling constant renormal-ization is then obtained in the standard way by requiringthat Y ( k F , be invariant under an infinitesimal reduc-tion of the running band cutoff D . Note that the bandcutoff appears in both, the intermediate electron propa-gator G c and in χ c . However, differentiation of the latterdoes not contribute to the logarithmic RG flow. Thisleads to the one-loop RG equation [24] dgd ln D = − g − y g D T K p D/T K ) ! , (5)where we have introduced the dimensionless couplings g = N (0) J , g = N (0) J , and D is the bare band cut-off. The first term on the right-hand side of Eq. (5) is theon-site contribution to the differential coupling renormal-ization (the β function), while the second term representsthe RKKY contribution. It is seen that χ f , as in Eq. (3),induces a soft cutoff on the scale T K and the characteris-tic /T K dependence to the RG flow of this contribution,where T K is the Kondo scale on the surrounding Kondosites. The dimensionless coefficient y = − Wπ Im X j = i e − i k F r ij N (0) G Rc ( r ij , Ω = 0) χ c ( r ij , Ω = 0) (6)arises from the Fourier transform Y ( k F , andparametrizes the RKKY coupling strength. The sum-mation in Eq. (6) runs over all positions j = i of Kondosites in the system. It is important to note that y isgenerically positive [24], even though the RKKY corre-lations χ c ( r ij , may be ferro- or antiferromagnetic. Forinstance, for an isotropic and dense system with latticeconstant a ( k F a ≪ ), the summation in Eq. (6) can beapproximated by an integral, and with the substitution x = 2 k F | r ij | , y can be expressed as y ≈ W ( k F a ) Z ∞ k F a dx (1 − cos x ) x cos x − sin xx > . (7)As a consequence, the RKKY correlations reduce the g − renormalization in Eq. (5), irrespective of the sign of χ c ( r ij , , as one would physically expect. Universal suppression of the Kondo scale. – The RG(5) can be integrated analytically [24]. The Kondo scalefor singlet formation on site i is defined as the runningcutoff value where the c − f coupling g diverges. Byequivalence of all Kondo sites, this is equal to the Kondoscale T K on the surrounding sites j = i , which appears asa parameter in the β function on the right-hand side ofEq. (5). This implies an implicit equation for the Kondoscale T K = T K ( y ) in a Kondo lattice, and that it dependson the RKKY parameter yT K ( y ) T K (0) = exp (cid:18) − y α g D T K ( y ) (cid:19) . (8)Here, T K (0) = D exp( − / g ) is the single-ion Kondoscale without RKKY coupling, and α = ln( √ .By the rescaling, u = T K ( y ) / ( yαg D ) , y c = T K (0) / ( αeg D ) , Eq. (8) takes the universal form ( e isEuler’s constant), yey c u = e − /u . (9)Its solution can be expressed in terms of the Lambert W function [28] as u ( y ) = − /W ( − y/ey c ) . The inset ofFig. 2 visualizes solving Eq. (9) graphically. It shows thatEq. (9) has solutions only for y ≤ y c . This means that y c marks a Kondo breakdown point beyond which the RGdoes not scale to strong coupling; i.e., a Kondo singlet isnot formed for y > y c even at the lowest energies. Us-ing the above definitions, the RKKY-induced suppressionof the Kondo lattice temperature reads T K ( y ) /T K (0) = u ( y ) y/ ( ey c ) = − y/ [ ey c W ( − y/ey c )] . It is shown in Fig. 2.In particular, at the breakdown point it vanishes discon-tinuously and takes the finite, universal value (see insetof Fig. 2), T K ( y c ) T K (0) = 1 e ≈ . . (10)We emphasize that the RKKY parameter y depends ondetails of the conduction band structure, including bandrenormalizations caused by the Kondo effect (coupling tothe heavy-fermion band). It also depends on the spatialarrangement of Kondo sites. Subleading contributionsto Γ cf may modify the form of the cutoff function in theRG (5) and thus the nonuniversal parameter α . However,all this does not affect the universal dependence of T K ( y ) on y given by Eq. (9).The critical RKKY parameter, as defined beforeEq. (9), can be expressed solely in terms of the single-ion Kondo scale y c = 4 αe τ K (ln τ K ) (11)with τ K = T K (0) /D . Note that [via T K (0) = D exp( − / g ) and N (0) = 1 / (2 D ) ] this is equivalentto Doniach’s breakdown criterion [6], N (0) y c J = T K (0) up to a factor of O (1) . However, the present theory goesbeyond the Doniach scenario in that it predicts the be-havior of T K ( y ) . pea k w i d t h [ m e V ] c FIG. 3: Comparison of the theory (9) (red curve) with STMspectroscopy experiments on a tunable two-impurity Kondosystem [18] (data points). The data points represent theKondo scale T K as extracted from the STM spectra by fit-ting a split Fano line shape of width T K to the experimentalspectra. Blue points: STM tunneling regime. Green points:contact regime. See Ref. [18] for experimental details. Comparison with experiments. – The present theoryapplies directly to two-impurity Kondo systems and canbe compared to corresponding STM experiments [18, 19].In Ref. [18], the Kondo scale has been extracted as theline width of the (hybridization-split) Kondo-Fano reso-nance. In this experimental setup, the RKKY parame-ter y is proportional to the overlap of tip and surface c electron wave functions and, thus, depends exponentiallyon the tip-surface separation z , y = y c exp[ − ( z − z ) /ξ ] .Identifying the experimentally observed breakdown point z = z with the Kondo breakdown point, the only ad-justable parameters are a scale factor ξ of the z coordi-nate and T K (0) , which is the resonance width at largeseparation, z = 300 pm. The agreement between theoryand experiment is striking, as shown in Fig. 3. In partic-ular, at the breakdown point T K ( y c ) /T K (0) coincides ac-curately with the prediction (10) without any adjustableparameter. In the STM experiment of Ref. [19], thestrongest observed suppression ratio is T K ( y ) /T K (0) =46 K /
110 K ≈ . , again in excellent agreement withthe strongest theoretical suppression of /e , consideringthat in Ref. [19] the RKKY coupling y cannot be variedcontinuously. The detailed analysis of that experimentwill be published elsewhere [29]. Discussion and conclusion. – We have derived a per-turbative renormalization group theory for the interfer-ence of Kondo singlet formation and RKKY interactionin Kondo lattice and multi-impurity systems, assumingthat magnetic ordering is suppressed, e.g., by frustration.The equivalence of the c − f vertices on all Kondo sites isreminiscent of a dynamical mean-field theory treatment;however, it goes beyond the latter in taking the nonlocalRKKY contributions into account. Equations (8) or (9)represent a mathematical definition of the energy scalefor Kondo singlet formation in a Kondo lattice, i.e., ofthe Kondo lattice temperature T K ( y ) . The theory pre-dicts a universal suppression of T K ( y ) and a breakdown ofcomplete Kondo screening at a critical RKKY parameter y = y c . At the breakdown point, the Kondo scale takesa finite , universal value, T K ( y c ) /T K (0) = 1 /e ≈ . ,and vanishes discontinuously for y > y c . In the Ander-son lattice, by contrast to the Kondo lattice, the localityof the f spin no longer strictly holds, but our approachshould still be valid in this case. The parameter-free,quantitative agreement of this behavior with differentspectroscopic experiments [18, 19] strongly supports thatthe present theory captures the essential physics of theKondo-RKKY interplay.The results may have profound relevance for heavy-fermion magnetic QPTs. In an unfrustrated lattice, thepartially screened local moments existing for y > y c mustundergo a second-order magnetic ordering transition atsufficiently low temperature. This will also imply a powerlaw divergence of the c electron correlation χ c in Eq. (2).We have checked the effect of such a magnetic instability,induced either by the ordering of remanent local momentsor by a c − electron SDW instability: the breakdown ratio T K ( y c ) /T K (0) will be altered, but must remain nonzero.The reason is that the inflection point of the exponentialfunction on the right-hand side of Eq. (9) (see Fig. 2)is not changed by such a divergence and, therefore, thesolution ceases to exist at a finite value of T K ( y c ) . Thispoints to an important conjecture about a possible, newquantum critical scenario with Kondo destruction: theKondo spectral weight may vanish continuously at theQCP, while the Kondo scale T K ( y ) (resonance width) re-mains finite, both as observed experimentally in Ref. [18].Such a scenario may reconcile apparently contradictoryexperimental results in that it may fulfill dynamical scal-ing, even though T K ( y c ) is finite at the QCP. 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Nejati, PhD thesis, University of Bonn, Germany,2016.[26] J. Kroha and P. Wölfle, Fermi and non-Fermi liquid be-havior in quantum impurity systems: conserving slave bo-son theory, Acta Phys. Polonica B , 3781 (1998). [27] N. Andrei, K. Furuya, and J. H. Lowenstein, Solution ofthe Kondo problem, Rev. Mod. Phys. , 331 (1983).[28] D. Veberić, Lambert W function for applications inphysics, Comput. Phys. Commun. , 2622 (2012);arXiv:1209.0735.[29] A. Nejati and J. Kroha, Oscillation and suppression ofKondo temperature by RKKY coupling in two-site Kondosystems, arXiv:1612.06620 [J. Phys.: Conf. Ser. (to bepublished)]. Supplemental Material for“Kondo destruction in RKKY-coupled Kondo lattice and multi-impurity systems”
Ammar Nejati, Katinka Ballmann, Johann Kroha , Physikalisches Institut and Bethe Center for Theoretical Physics,Universität Bonn, Nussallee 12, D-53115 Bonn, Germany Center for Correlated Matter, Zhejiang University, Hangzhou, Zhejiang 310058, China (Dated: December 8, 2016)In this supplement we provide details of (1) the calculation of the spin scattering vertex between local f − spinsand conduction electrons, including RKKY contributions and (2) the derivation of the one-loop RG equation and itssolution. I. F-SPIN – CONDUCTION ELECTRONVERTEX ˆΓ cf The elementary f − spin – c − electron vertex with cou-pling constant J is defined via the Kondo Hamiltonian, H = X k ,σ ε k c † k σ c k σ + J X i S i · s i , (S1)with the notation used in the main text. The direct ( d )and exchange ( x ) parts of the RKKY-induced vertex canbe written as the product of a distance and energy de-pendent function Λ ( d/x ) RKKY and an operator in spin space, ˆ γ ( d/x ) , ˆ γ ( d/x ) RKKY = Λ ( d/x ) RKKY ( r ij , i Ω) ˆ γ ( d/x ) (S2) A. Spin structure
Denoting the vector of Pauli matrices acting in c − electron spin space by σ = ( σ x , σ y , σ z ) and the vectorof Pauli matrices in f − spin space by s = ( s x , s y , s z ) , theRRKY-induced vertex contributions read in spin space, γ ( d ) αβ,κλ = X a,b,c = x,y,z X γ,δ,µ,ν =1 (cid:0) σ aδγ s aκλ (cid:1) (cid:0) σ bγδ s bνµ (cid:1) (cid:0) σ cαβ s cµν (cid:1) (S3) γ ( x ) αβ,κλ = X a,b,c = x,y,z X γ,δ,µ,ν =1 (cid:0) σ aδγ s aκλ (cid:1) (cid:0) σ bαδ s bνµ (cid:1) (cid:0) σ cγβ s cµν (cid:1) (S4)with c − electron spin indices α , β , γ , δ , and f − spinindices κ , λ , µ , ν , as shown in Fig. S1. The spinsummations can be performed using the spin algebra( a, b = x, y, z ), X γ =1 σ aαγ σ bγβ = X c = x,y,z iε abc σ cαβ + δ ab αβ , (S5) χ f jji RKKY γ f χ jj i RKKY(x) γ νµ αβ δλ κγ (d) λβ αγ δµν κ FIG. S1: Direct ( d ) and exchange ( x ) diagrams of the RKKY-induced contributions to the c − f vertex: spin labelling. where is the unit operator in spin space, ε abc the to-tally antisymmetric tensor and δ ab the Kronecker- δ . Thisresults in a nonlocal Heisenberg coupling between sites i and j , γ ( d ) αβ,κλ = 4 X a = x,y,z (cid:0) σ aαβ s aκλ (cid:1) (S6) γ ( x ) αβ,κλ = − X a = x,y,z (cid:0) σ aαβ s aκλ (cid:1) . (S7) B. Energy dependence
With the energy variables as defined in Fig. S2, the en-ergy dependent functions in Eq. (S2) read in Matsubararepresentation (the Matsubara indices are suppressed forsimplicity of notation, i.e. i Ω ≡ i Ω n , etc.), f χ jj i χ f jji ω+Ωω RKKY γ RKKY(x) γ (d) ε+Ωε Ω εω ω+Ωε+ω ε+ω+Ω FIG. S2: Direct ( d ) and exchange ( x ) diagrams of the RKKY-induced contributions to the c − f vertex: energy labelling. Λ ( d ) RKKY ( r ij , i Ω) = JJ χ c ( r ij , i Ω) ˜ χ f ( i Ω) (S8) Λ ( x ) RKKY ( r ij , iω, i Ω) = − JJ T X iε G c ( r ij , iω + iε ) G c ( r ij , iω + i Ω + iε ) ˜ χ f ( iε ) (S9)where χ c ( r ij , i Ω) = − T X ε G c ( r ij , iε ) G c ( r ij , iε + i Ω) (S10)and ˜ χ f ( iε ) = χ f ( iε ) / ( g L µ B ) , with χ f ( iε ) the full, single-impurity f − spin susceptibility whose temperature depen-dence is known from Bethe ansatz (see main text). In an isotropic system in d = 3 dimensions, the retarded conductionelectron Green’s function G c as well as the susceptibilities χ c and ˜ χ f at temperature T = 0 are calculated in positionspace as, G c ( r , ω ± i
0) = − πN ( ω ) e ± ik ( ε F + ω ) r k ( ε F + ω ) r (S11) χ c ( r ij , Ω + i
0) = " N (0) sin( x ) − x cos( x )4 x + O (cid:18) Ω ε F (cid:19) ! ± i " π N (0) 1 − cos( x ) x Ω ε F + O (cid:18) Ω ε F (cid:19) ! (S12) ˜ χ f (Ω ± i
0) =
WπD D T K p /T K ) (cid:18) ± iπ arsinh Ω T K (cid:19) with A := WπD . (S13)Here, ε F and k F are the Fermi energy and Fermi wavenumber, respectively, x = 2 k F r , N (0) the conduction electrondensity of states at the Fermi energy, and D the bare band cutoff.For the renormalization of the total c − f vertex for c − electrons at the Fermi energy, the contributions Λ ( d ) RKKY , Λ ( d ) RKKY must be calculated for real frequencies, i Ω → Ω + i , iω → ω + i , and for electrons at the Fermi energy,i.e., ω = 0 . In this limit, only the real parts of Λ ( d ) RKKY , Λ ( d ) RKKY contribute to the vertex renormalization, as shownbelow. In order to analyze their importance for the RG flow, we will expand them in terms of the small parameter T K /D . In the following, the real part of a complex function will be denoted by a prime ’ and the imaginary part bya double-prime ”. Direct contribution
Since in Λ ( d ) RKKY [Eq. (S8)], χ c ( i Ω) and ˜ χ f ( i Ω) appear as a product and χ f (Ω) cuts off the energy transfer Ω at thescale T K ≪ ε F ≈ D , the electron polarization χ c (Ω) contributes only in the limit Ω ≪ ε F where it is real-valued,as seen in Eq. (S12). Using Eq. (S12) and Eq. (S13), the real part of the direct RKKY-induced vertex contributionreads, Λ ( d ) RKKY ′ ( r ij , Ω + i
0) = JJ R ( r ij ) AN (0) D T K p /T K ) + O (cid:18) Ω D (cid:19) ! , (S14)where R ( r ij ) = sin( x ) − x cos( x )4 x , x = 2 k F r (S15)is a spatially oscillating function. Exchange contribution
In order to analyze the size of Λ ( x ) RKKY ′ in terms of T K /D , it is sufficient to evaluate it for a particle-hole symmetricconduction band and for r ij = 0 , since the T K /D dependence is induced by the on-site susceptibility ˜ χ f ( i Ω) . Thedependence on T K /D can be changed by the frequency convolution involved in Λ ( x ) RKKY ′ , but does not dependon details of the conduction band and distance dependent terms. (The general calculation is possible as well, butconsiderably more lengthy [2].) We use the short-hand notation for the momentum-integrated c − electron Green’sfunction, G c ( r = 0 , ω ± i
0) = G ( ω ) = G ′ ( ω ) + iG ′′ ( ω ) , and assume a flat density of states N ( ω ) , with the upper andlower band cutoff symmetric about ε F , i.e., G R/A ′′ ( ω ) = ∓ π D Θ( D − | ω | ) (S16) G R/A ′ ( ω ) = 12 D ln (cid:12)(cid:12)(cid:12)(cid:12) D + ωD − ω (cid:12)(cid:12)(cid:12)(cid:12) = ωD + O (cid:18)(cid:18) ωD (cid:19)(cid:19) . (S17)Furthermore, at T = 0 the Fermi and Bose distribution functions are, f ( ε ) = − b ( ε ) = Θ( − ε ) . Λ ( x ) RKKY ′ (0 , , Ω + i then reads, Λ ( x ) RKKY ′ ( r ij = 0 , ω = 0 + i , Ω + i
0) = − JJ (cid:26) Z dεπ (cid:2) f ( ε ) G A ′′ ( ε ) G R ′ ( ε + Ω) + f ( ε + Ω) G A ′ ( ε ) G A ′′ ( ε + Ω) (cid:3) ˜ χ Rf ′ ( ε ) (S18) − Z dεπ (cid:2) f ( ε ) G R ′ ( ε ) G R ′ ( ε + Ω) − f ( ε + Ω) G A ′′ ( ε ) G A ′′ ( ε + Ω) (cid:3) ˜ χ Rf ′′ ( ε ) (cid:27) . With the above definitions, the four terms in this expression are evaluated in an elementary way, using the substitution ε F /T K = x = sinh u , Z dεπ f ( ε ) G A ′′ ( ε ) G R ′ ( ε + Ω) ˜ χ Rf ′ ( ε ) = AN (0) T K D − s (cid:18) D T K (cid:19) + Ω T K arsinh (cid:18) D T K (cid:19) = AN (0) (cid:20) − D ln (cid:18) D T K (cid:19) + O (cid:18) T K D (cid:19)(cid:21) (S19) (cid:12)(cid:12)(cid:12)(cid:12)Z dεπ f ( ε + Ω) G A ′ ( ε ) G A ′′ ( ε + Ω) ˜ χ Rf ′ ( ε ) (cid:12)(cid:12)(cid:12)(cid:12) = AN (0) T K D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)s (cid:18) Ω T K (cid:19) − s (cid:18) D + Ω T K (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ AN (0) + O (cid:18) T K D (cid:19) (S20) Z dεπ f ( ε ) G R ′ ( ε ) G R ′ ( ε + Ω) ˜ χ Rf ′′ ( ε ) = − π AN (0) (cid:18)
12 + Ω D (cid:19) ln (cid:18) D T K (cid:19) + O (cid:18) T K D (cid:19) ! (S21) Z dεπ f ( ε + Ω) G A ′′ ( ε ) G A ′′ ( ε + Ω) ˜ χ Rf ′′ ( ε ) = π AN (0) (cid:20) − arsinh (cid:18) Ω T K (cid:19) + arsinh (cid:18) min (cid:18) Ω T K , D + Ω T K (cid:19)(cid:19)(cid:21) ≤ π AN (0) + O (cid:18) T K D (cid:19) . (S22)0Comparing Eqs. (S18)–(S22) with Eq. (S14) shows that all terms of Λ ( x ) RKKY ′ (Ω) are subleading compared to Λ ( d ) RKKY ′ (Ω) by at least a factor ( T K /D ) ln( T K /D ) for all transfered energies Ω . Hence, it can be neglected in theRG flow. Combining the results of spin and energy dependence, Eqs. (S2), (S6) and (S14), one obtains the totalRKKY-induced c − f vertex as, ˆ γ ( d ) RKKY ( r ij , i Ω) = 2 (1 − δ ij ) χ c ( r ij , i Ω) χ f ( i Ω) S i · s j (S23)or Reˆ γ ( d ) RKKY ( r ij , Ω + i
0) = 2 JJ AN (0) (1 − δ ij ) R ( r ij ) D T K p /T K ) S i · s j . (S24) II. PERTURBATIVE RENORMALIZATION GROUPA. One-loop RG equation
The derivation of the RG equation follows the well-known procedure of perturbative coupling constant renormaliza-tion [1], however performed for the nonlocal c − f vertex including RKKY contributions. The amplitude of the sumof the diagrams contributing to the one-loop renormalization of the c − f vertex reads in Matsubara representation(c.f. Fig. 1 (b) and Eq. (4) of the main paper), Y ( r ij , iω ) = − T X i Ω h J δ ij + Jγ ( d ) RKKY ( r ij , i Ω) + Jγ ( d ) RKKY ( r ij , − i Ω) i G c ( r ij , iω − i Ω) G f ( i Ω) (S25) + T X i Ω h J δ ij + Jγ ( d ) RKKY ( r ij , i Ω) + Jγ ( d ) RKKY ( r ij , − i Ω) i G c ( r ij , iω + i Ω) G f ( i Ω) This is a nonlocal function of the ingoing and outgoing coordinates of c − electrons, x j , x i . For the low-energy strongcoupling fixed point the coupling constant for c − electrons at the Fermi energy must be renormalized, i.e., for excitationenergy ω = 0 and momentum on the Fermi surface, k F . Hence, the coupling constant renormalization is given by theFourier transform of Y ( r ij , ω = 0 + i with respect to the in- and outgoing coordinates, x j , x i , taken for momenta k in , k out on the Fermi surface. In a lattice system, translation invariance implies the conservation of in in- andoutgoing momenta, k in = k out = k F . This yields, Y ( k F , iω ) = − J T X i Ω [ G c ( r ij → , iω − i Ω) G f ( i Ω) − G c ( r ij → , iω + i Ω) G f ( i Ω)] (S26) − J T X i Ω X j e + i k F r ij h γ ( d ) RKKY ( r ij , i Ω) + γ ( d ) RKKY ( r ij , − i Ω) i G c ( r ij , iω − i Ω) G f ( i Ω)+
J T X i Ω X j e − i k F r ij h γ ( d ) RKKY ( r ij , i Ω) + γ ( d ) RKKY ( r ij , − i Ω) i G c ( r ij , iω + i Ω) G f ( i Ω) , where the j − summation runs over all lattice sites j which are occupied by a Kondo ion. Using the symmetry γ ( d ) RKKY ( r ij , − Ω) = γ ( d ) ∗ RKKY ( r ij , Ω) , and the pseudofermion propagator G f ( i Ω) = 1 /i Ω , projected onto the physicalHilbert space [3], the frequency summations can be performed to yield, Y ( k F , ω = 0) = − J N (0) "Z DT d ΩΩ − Z − T − D d ΩΩ (S27) − J π Z DT d ΩΩ Im X j e + i k F r ij γ ( d ) RKKY ′ ( r ij , Ω) G c ( r ij , Ω − i Where D is the running band cutoff. The change of Y ( k F , ω = 0) under an infinitesimal, logarithmic cutoff reduction d ln D represents the renormalization of the c − f coupling constant. That is, the renormalization group equation isobtained as, dgd ln D = dY ( k F , ω = 0) N (0) d ln D = − g − π g Im X j e − i k F r ij γ ( d ) RKKY ′ ( r ij , D ) G c ( r ij , D + i , (S28)1 x x -0.00400.004 FIG. S3: Integrand of the expression for y , Eq. (S31). Itillustrates that y > and that y depends sensitively on k F a .The inset shows the curve on a smaller scale. where g = JN (0) , g = J N (0) are the dimensionlesscouplings. Defining the RKKY coupling parameter as, y = − Wπ Im X j = i e − i k F r ij N (0) G Rc ( r ij , Ω = 0) χ c ( r ij , Ω = 0) , (S29)the RG equation takes the form, dgd ln D = − g " − yg D T K p /T K ) . (S30)It naturally reduces to the single-impurity Kondo RGequation, if the RKKY interaction is switched off ( y = 0 ).In a dense Kondo lattice with lattice constant a in d = 3 dimensions, k F a ≪ , the lattice summation in Eq. (S29)can be approximated by an integral, and y becomes, y ≈ W ( k F a ) Z ∞ k F a dx (1 − cos x ) x cos x − sin xx > . (S31) y depends sensitively on k F a . It represents the depen-dence on the lattice structure of the Kondo ions ( a ) andon the band filling ( k f ). For illustration we show inFig. S3 the integrand of the expression for y , Eq. (S31).It is also seen that y > , i.e. the RKKY couplingalways reduces the effective coupling strength betweenconduction electrons and local f − spins, irrespective ofthe oscillating sign of the RKKY correlations, Eq. (S12).This is physically expected, since a ferro- as well asan antiferromagnetic coupling of an f − spin to the neighboring f − spins will always reduce the local spinfluctuations and, therefore, the Kondo singlet formation.For a non-translational invariant systems, like two- ormulti-impurity Kondo systems, the expression for y willsomewhat differ from Eq. (S29), since in- and outgoingmomenta are not conserved. However, this does notchange the form of the RG equation (S30). B. Integration of the RG equation
The RG equation Eq. (S30) is readily integrated byseparation of variables, − Z gg dgg = 2 Z ln D ln D d ln D ′ (S32) − yg D T K Z D/T K D /T K dxx √ x , or g − g = 2 ln (cid:18) DD (cid:19) (S33) − yg D T K ln p D/T K ) − p D/T K ) + 1 ! where we have used D /T K ≫ in the last expression.The Kondo scale is defined as the value of the run-ning cutoff D where g diverges, i.e., g → ∞ when D → T K . This yields the defining equation for the Kondoscale which, hence, depends on the RKKY parameter, T K ≡ T K ( y ) , − g = 2 ln (cid:18) T K ( y ) D (cid:19) (S34) − yg D T K ( y ) ln √ − √ ! Using the definition of the single-impurity Kondo tem-perature, − /g = 2 ln ( T K (0) /D ) , the defining equa-tion for T K ( y ) can finally be written as T K ( y ) T K (0) = exp (cid:18) − yαg D T K ( y ) (cid:19) , (S35)with α = ln( √ . [1] A. C. Hewson, The Kondo Problem to Heavy Fermions,Cambridge University Press , Cambridge, UK (1993).[2] A. Nejati,
Quantum phase transitions in multi-impurityand lattice Kondo systems , PhD thesis, University ofBonn, Germany (2016). [3] J. Kroha and P. Wölfle, Fermi and non-Fermi liquid be-havior in quantum impurity systems: conserving slaveboson theory, Acta Phys. Polonica B29