Local moment dynamics and screening effects in doped charge-transfer insulators
LLocal moment dynamics and screening effectsin doped charge-transfer insulators
A. Amaricci, N. Parragh, M. Capone, and G. Sangiovanni Democritos National Simulation Center, Consiglio Nazionale delle Ricerche,Istituto Officina dei Materiali (IOM) and Scuola Internazionale Superioredi Studi Avanzati (SISSA), Via Bonomea 265, 34136 Trieste, Italy Institute for Theoretical Physics and Astrophysics,University of W¨urzburg, D-97074 W¨urzburg, Germany (Dated: April 4, 2019)By means of Dynamical Mean-Field Theory we investigate the spin response function of a modelfor correlated materials with d - or f -electrons hybridized with more delocalized ligand orbitals.We point out the existence of two different processes responsible for the dynamical screening oflocal moments of the correlated electrons. Studying the local spin susceptibility we identify thecontribution of the “direct” magnetic exchange and of an “indirect” one mediated by the itinerantuncorrelated orbitals. In addition, we characterize the nature of the dynamical screening processesin terms of different classes of diagrams in the hybridization-expansion contributing to the density-matrix. Our analysis suggests possible ways of estimating the relative importance of these twoclasses of screening processes in realistic calculations for correlated materials. PACS numbers: 71.27.+a, 71.10.Fd, 71.30.+h
I. INTRODUCTION
The physics of strongly correlated systems can be iden-tified with the process that turns delocalized electronsinto localized magnetic moments. In d - and f -electronmaterials, such as transition-metal oxides (TMO) andheavy fermions (HF), the confined nature of the corre-lated orbitals gives rise to large spin moments localizedover short time scales at each lattice site. In the atomiclimit, in which the system is described as a collection ofdisconnected atoms, such local spins assume the largestvalue allowed by the atomic configuration, and show nodynamics. On the other hand, in the presence of a frac-tion of itinerant electrons ( i.e. away from the extremecase of the atomic limit), the instantaneous value of thelocal spins gets reduced and a dynamics emerges as aneffect of the screening processes.In TMO sizeable local magnetic moments from d -orbital electrons have been revealed by inelastic neu-tron scattering (INS) and X-ray absorption spectroscopy(RIXS) in materials like, e.g. high-T c cuprates ,cobaltate and iron-based superconductors . The fin-gerprints of the individual screening processes are hardto extract from the instantaneous value of the local mo-ments. For instance, only indirect information aboutthe nature of the screening of the local moments can begained from the temperature dependence of the local spinsusceptibility . The spin dynamics can instead be amuch more sensitive tool to diagnose the physical effectsof such screening processes .From a general point of view one expects two processesto be responsible for the screening of the local moments: i ) processes involving a direct hopping between correlatedelectrons, and ii ) processes involving the hybridization with more itinerant, e.g. ligand p -, orbitals. In the ox-ides, one associates processes of type i ) with the super- exchange mechanism which is captured by a Hubbard-like description including only correlated d -orbitals. This“ d -only” description is good whenever the d -manifold isvery well isolated from the ligand p -bands. Processes oftype ii ) can instead be viewed as an additional screen-ing channel, active if the system can gain delocalizationenergy upon hybridizing with the bath of itinerant elec-trons.In the present paper we study the local moment dy-namics and the screening effects addressing questionssuch as: “how can we distinguish between type- i ) andtype- ii ) contributions to local spin susceptibility?”, or“which features would we expect to see in a INS or RIXSexperiment if the screening is dominated by hybridiza-tion processes?” In order to do this we focus our at-tention on doped “charge-transfer” insulators (CTI), e.g. Cuprates, as a paradigmatic example of Mott sys-tems in which the effects of the hybridization are crucial.We therefore consider a generic, yet simple, model fora CTI in which the doping and the relative importancebetween the two screening channels can be easily tuned.By using Dynamical Mean-Field Theory (DMFT) westudy the local dynamical spin response function. Wepoint out the existence of two distinct features in the lo-cal spin susceptibility associated to processes of the twodifferent types and we numerically characterize their na-ture in a clear way in terms of different classes of dia-grams contributing to the density-matrix.The structure of the paper is as follow: in Sec. II weintroduce the theoretical model and briefly discuss itsnumerical solutions within DMFT. In Sec. III we discussthe results for the local moment dynamics. In Sec. IVwe characterize the different features in the spin sus-ceptibility in terms of distinct class of diagrams in thestrong-coupling expansion. Finally, section V containsconcluding remarks. a r X i v : . [ c ond - m a t . s t r- e l ] O c t II. MODEL
We consider a generalized periodic Anderson model( t dd -PAM ) describing a wide-band of conduction elec-trons, hybridizing with a narrow-band of strongly inter-acting electrons: H = (cid:88) k σ ε p ( k ) p + k σ p k σ + (cid:88) k σ ε d ( k ) d + k σ d k σ + t pd (cid:88) iσ (cid:0) d + iσ p iσ + p + iσ d iσ (cid:1) + U (cid:88) i d + i ↑ d + i ↓ d i ↑ d i ↓ (1)The operators p iσ ( p + iσ ) destruct (create) electrons inthe conduction band with spin- σ with dispersion ε p ( k ) = (cid:15) p − t pp [cos( k x ) + cos( k y )]. Similarly, d iσ ( d + iσ ) destruct(create) electrons in the narrow-band with spin- σ anddispersion ε d ( k ) = (cid:15) d − αt pp [cos( k x ) + cos( k y )], where α ∈ [0 ,
1) denotes the bandwidth ratio. The two or-bital electrons hybridize with a local amplitude t pd . Thelast term in Hamiltonian (1) indicates the strong localCoulomb interaction U experienced by the d -electrons.In the following, we fix the energy unit to the half-bandwidth of the conduction electrons D = 4 t pp = 1. Inaddition we shall set the bandwidth ratio to α = 0 . (cid:15) d = 0. The energy separation between the centersof the two bands ∆ = (cid:15) p − (cid:15) d denotes the charge-transferenergy. Finally, we will drop any reference to the spinindex, as we focus on the paramagnetic state, where thelocal moments are not ordered.The model Hamiltonian (1) interpolates between theHubbard model (HM) for the correlated d -electrons( t pd = 0, α (cid:54) = 0), and the more usual periodic Ander-son model ( α = 0, t pd (cid:54) = 0), describing non-dispersivecorrelated electrons hybridized with a wide-band .We solve the t dd -PAM using Dynamical Mean FieldTheory (DMFT) . The DMFT allows us to study thelocal screening and the spin dynamics in a fully non-perturbative way. Within DMFT, the lattice model (1)is mapped onto an effective impurity problem for a sin-gle d -orbital, supplemented by a self-consistency condi-tion for the local Weiss Field (WF). The WF G − dd iscalculated by isolating the dd -element of the interact-ing local Green’s function, as in general DMFT schemeswith enlarged basis-sets . We solve the associatedeffective impurity problem using exact-diagonalization(ED) and hybridization-expansion continuous-timequantum Monte Carlo (CTQMC) methods .Within ED the local WF must be represented in termsof a discretized hybridization function: ∆ dd ( iω n ) = (cid:80) N b l =1 V l / ( iω n − ε l ) using N b auxiliary energy levels.The parameters ε l and V l describe, respectively, the localenergy and the hybridization between the impurity andthe l th bath level. All the ED calculations are performedusing N b = 6.The spin susceptibility χ spin is defined as the imaginarypart of the dynamical response function: χ spin ( ω ) = i (cid:90) dte iωt θ ( t ) (cid:104) [ ˆ S ( d ) z ( t ) , ˆ S ( d ) z (0)] (cid:105) (2) U [ + t pd ] b /20.50.60.70.80.91 t pd =0.25t pd =0.75 Hubbard - d =0.10 t pd =0.0625 t pd =0.125 t pd =0.25 t pd =0.50 t pd =0.75 t pd =1.00 Hubbard - d =0.10 Figure 1. (Color online) Spin susceptibility χ (cid:48)(cid:48) spin ( ω ) on thereal-axis for increasing values of t pd . The different curves areshifted along the y -axis by 2 t pd for better comparison. Dataare for doping value δ = 0 .
10. Dotted line is the shift inthe position of the p -band from increasing hybridized bandrepulsion. Inset: imaginary-time spin susceptibility χ (cid:48)(cid:48) spin ( τ )for a fixed doping δ = 0 .
10 for t pd = 0 .
25 and 0 .
75. The blackcurve corresponds to the Hubbard model result with doping δ = 0 . where ˆ S ( d ) z is the z -component of the spin operator onthe d -site and [ , ] denotes the commutator. In ED thespin susceptibility is evaluated using the spectral decom-position: χ (cid:48)(cid:48) spin ( ω ) = π Z (cid:88) i,j |(cid:104) i | ˆ S ( d ) z | j (cid:105)| ( e − βE j + − e − βE i ) δ ( ω − ( E i − E j )) , (3)where Z is the partition function.The hybridization-expansion CTQMC method pro-vides a (statistically) exact solution of the DMFT equa-tions. Indeed, we tested the agreement between ED andCTQMC calculations finding very satisfactory results forboth local and dynamical quantities. As we will show,CTQMC permits to investigate the diagrams contribut-ing to the local screening processes. Since this is doneusing a (infinite series) perturbation-expansion language,it turns out to give useful information about the physicsinvolved in the screening of the local moment. III. SPIN SUSCEPTIBILITY
As discussed in the Introduction, we focus on dopedCTI. By definition this means that the hole doping in-volves mainly the p -band. This marks a strong differencewith a description of pure d -electrons, where the insu-lator has a Mott-Hubbard character. In order to studythe differences in the spin susceptibility and in other ob-servables induced by the hybridization t pd , we want to d =0.10 t pd =0.25 t pd =0.50 t pd =0.75 t pd =1.00 0.1 1 t pd d =0.05 d =0.10 d =0.20 Hubbard - d =0.10 Hubbard (a) (c)(b)
Figure 2. (Color online) (a) Imaginary part of the Matsubara d -electron self-energy ImΣ dd for increasing hybridization am-plitude t pd and doping δ = 0 .
10. (b) p - and d -orbital densities, (cid:104) n p (cid:105) and (cid:104) n d (cid:105) respectively, as a function of t pd . The arrowto filled symbol indicates (cid:104) n d (cid:105) for the Hubbard model withdoping δ = 0 .
10. (c) d -electron local moment m d = (cid:104) ˆ S ( d ) z (cid:105) as a function of t pd and different values of the doping δ . Thearrows indicate the Hubbard model results at the same valuesof δ . be able to compare solutions with both finite t pd and fi-nite doping to a doped Hubbard model. However, we cannot recover the latter in the limit of vanishing t pd of ourmodel, as this tends towards a half-filled , “ d -only” Mottinsulator. Hence, we shall complement our calculationsby solving the Hubbard model for a given (in principlearbitrary) value of the hole-doping δ . We will fix δ ac-cording to physically motivated criteria, e.g. that the sizeof the instantaneous spin moment is that of the solutionwith finite t pd we are comparing to, or, that the occu-pation of the d -orbital (cid:104) n d (cid:105) is the same between the twomodels.In order to place the system into the “charge-transfer”regime we consider the model defined in Eq. (1) with U =3 . − .
5. In addition we set the temperatureto T = 1 / χ spin (both in ω and in τ )are shown for different values of t pd and total occupation n = (cid:104) n d (cid:105) + (cid:104) n p (cid:105) = 2 .
9, as well as for the “ d -only” Hubbardmodel at δ = 0 .
10. As it can be seen in the inset ofFig.1 and in Fig.2c, where we show the d -electrons localmoment m d = (cid:104) ˆ S ( d ) z (cid:105) , the latter calculation yields thesame instantaneous ( i.e. τ = 0) moment of the case ofthe t dd -PAM with t pd = 0 . χ spin ( ω )of Fig.1 in more detail. In the “ d -only” Hubbard casethe instantaneous local moment is dynamically screened Hubbard - ( t dd -PAM )Hubbard - d =0.10 t dd -PAM Figure 3. (Color online) Comparison of the spin susceptibility χ spin ( ω ) of the t dd -PAM with t pd = 0 .
75 and doping δ = 0 . d -only” Hubbard model calculations(thin and dashed lines). The thin line shows the case with d -electrons occupation (cid:104) n d (cid:105) equal to the value of n d in the t dd -PAM . The dashed line show the case at fixed doping δ = 0 .
10. Inset: blow-up of the low-frequency behavior fromthe main panel. by coherent metallic excitations at the Fermi level, asindicated by the vanishing imaginary part of the self-energy (see Fig.2a). In this regime, the screening processis entirely coming from d - d direct exchange with a lead-ing coupling J dd (cid:39) α t pp /U , as the p -electrons are com-pletely decoupled. Correspondingly, the spin suscepti-bility χ spin ( ω ) is dominated by low-energy contributions,though weaker high-energy features at ω (cid:39) U , associatedto electronic excitations across the Hubbard bands, canbe detected.If we now consider the extreme case of very large hy-bridization strength, i.e. t pd = 1, we notice pronouncedchanges in the spin susceptibility. First of all the low-frequency part acquires more structures, as further un-derlined in Fig.3 where the case t pd = 0 .
75 is directlycompared to two “ d -only” solutions: one for δ = 0 .
10 andthe other for the same value of (cid:104) n d (cid:105) as in our model.Big differences can be seen in the intermediate-to-highfrequency region. The χ spin ( ω ) of the t dd -PAM acquiresthere a significant weight and several additional peaksare visible. As highlighted in Fig.1, these features ex-tend in a frequency range with a width set by the p -electrons bandwidth 2 D , while their position scales with (cid:113) ∆ + 4 t pd . The latter corresponds to the correctionto the charge-transfer energy from hybridized bands re-pulsion, confirming that the intermediate-to-high lyingpeaks come from the hybridization with the more delo-calized p -orbitals.We now focus on the low-frequency region, i.e. for ω <
1. A blow-up is shown in the inset to Fig.3. Ev-idently, the two “ d -only” calculations both display lessstructures than the t dd -PAM (blue line). This suggests -3 -2 -1 0 1 2 30.1110 [ + . t pd ] t pd =1.00 t pd =0.75 t pd =0.50 t pd =0.25 t pd =0.125 t pd =0.0625 -2 0 2 4 00.010.020.030.040.05 Hubbard - d =0.10 Figure 4. (Color online) Spectral density of the hybridizationfunction ρ ( ω ) = − Im∆ dd ( ω ) /π for increasing amplitudes t pd .Inset: the same quantity for the “ d -only” Hubbard model. that the low-frequency feature of the spin susceptibility ofthe t dd -PAM has a mixed d - p character. In fact, in thisregion we expect both the d - d screening processes andthe d - p ones to be active. To estimate the order of thescreening processes one can consider that in the presenceof finite hybridization the d -electrons have an effectivehopping of the order t eff = αt pp + t pd / ∆. Then, usinga simple super-exchange argument, we can associate anumber of coupling constants to the different local mo-ments screening processes as follows: J dd (cid:39) α t pp U , J (1) pd (cid:39) αt pp ∆ U t pd , J (2) pd (cid:39) t pd ∆ U . (4)As pointed out before, the first constant describes di-rect d - d processes. The other two describe screeningprocesses involving two or four hybridizations with non-interacting electrons and, respectively, one or no directhopping events. For large values of t pd the J (1) pd dominatesat low-energy. On the other hand, for small value of thehybridizations J (2) pd becomes smaller than J (1) pd and the as-sociated exchanges processes dominates at low-frequency.We can attempt to relate the estimates of Eq. (4) to thestructures of χ spin ( ω ) shown in the inset to Fig.3. For theparameters used, J dd is the smallest coupling ( O (10 − ))and it can be associated to the lowest-energy onset of χ spin ( ω ) present in all three cases. J (1) pd assumes the valueof 0.04 which roughly corresponds to the position of thefirst deviation (dip) between the t dd -PAM curve and the“ d -only” Hubbard solutions. The largest coupling J (2) pd (of the order of 0.4) falls in the region separating thelow-energy structures from the intermediate-energy ones,where the largest deviations from the “ d -only” Hubbardstart to appear.In the remaining part of this section we discuss thebehaviour of the spin susceptibility as we decrease t pd down to very small values. As shown in Fig.2b this cor- t pd =0.25 t pd =0.50 t pd =0.75 t pd =1.00 t pd nd - p ea k po s iti on peak positionquadratic fit Hubbard - d =0.10 Figure 5. (Color online) Histogram h ( k ) of the distribution offermionic diagrams contributing to the local CTQMC tracefor different expansion order k . The histograms are shownfor different values of the hybridization t pd at fixed doping δ = 0 .
10. For the Hubbard model case (dashed line) the dis-tribution has one single peak. For the t dd -PAM ( t pd (cid:54) = 0) thecurves assume a bi-modal distribution with both “low-order”and “high-order” features. Each of these curves is fitted by adouble gaussian (dotted line). Inset: The position of the sec-ond peak scales quadratically in t pd . The peak position anderror bars are estimated from the mean value and standarddeviation of the double gaussian fits. responds to reducing the mixed valence character of the t dd -PAM solution. As we mentioned above, the occupa-tion (cid:104) n d (cid:105) of the d -orbital approaches 1 from above while (cid:104) n p (cid:105) saturates to 1.9 in order to keep the total densityto 2.9. Concomitantly, the size of the instantaneous mo-ment (see Fig.2c) increases towards the atomic value of1. The imaginary part of the self-energy becomes verylarge at low frequency, reflecting the strong incoherentcharacter of the solution (see Fig.2a). In this regimeof very small t pd the screening is poor. Indeed, the spinsusceptibility is very small and essentially featureless. Aremnant of the low-frequency peak can still be detectedfor t pd = 0 .
125 and a structure related to the excitationsbetween the lower Hubbard band and the (suppressed)spectral density at the Fermi level, is recognizable at en-ergies of order U . IV. DIAGRAMMATIC CHARACTERIZATION
The previous analysis revealed the existence of a num-ber of new features in the spin susceptibility which arean inevitable consequence of the inclusion of p -electrons.Yet, χ spin is not the ideal physical quantity to under-stand whether or not the new hybridization processescome entirely from p degrees of freedom. In this sectionwe introduce a quantity which turns out to be able to dis-criminate between d and p character of the hybridizationprocesses (i) (ii) (iii) (iv) Figure 6. (Color online) State-resolved density-matrix contribution to the expansion order histogram (see text). Data are forthe t dd -PAM with t pd = 0 . δ = 0 .
10 and bath spin-state σ = ↑ . The states of ˆ ρ α in different panels are | (cid:105) (i), |↑(cid:105) (ii), |↓(cid:105) (iii), |↑↓(cid:105) (iv). The figure shows the different contribution to the trace coming from the possible states configuration. The empty anddoubly occupied states do not contribute much to the trace. The only significant contribution comes from the singly occupiedstates and interestingly the two peaks have complementary character. To begin with, we consider the hybridization function∆ dd ( ω ) = (cid:80) N b l =1 V l / ( ω + − (cid:15) l ) on the real-frequency axis.This contains essential information about the formationof electronic excitations involved in the screening chan-nel. Then, using this discretized hybridization function,we perform strong-coupling CTQMC calculations in or-der to “visualize” distinct classes of diagrams responsiblefor the different screening effects.In Fig.4 we show the spectral density of the hybridiza-tion function ρ ( ω ) = − Im∆ dd ( ω ) /π for several values of t pd and finite doping. For the Hubbard model (see inset)this quantity has a finite weight at the Fermi level, sep-arated by higher energy feature describing hybridizationevents with doubly occupied states (Hubbard band). Forour model at tiny values of t pd ρ ( ω ) shows a dramaticreduction of the weight at Fermi level, in agreement withthe loss of coherence of this metallic state. It is very clearhow increasing t pd drives the formation of substantialspectral weight below and at the Fermi level. Thereforethe system gains a lot of kinetic energy by introducinghybridization events in that frequency region.We now turn our attention to the effects introduced bythese “new” hybridization events, from a diagrammaticpoint of view. In Fig.5 we show the fraction of diagramscontributing to the fermionic trace in the CTQMC cal-culation as a function of the expansion order, i.e. thehistogram h ( k ) of the order of the diagrams involved inthe calculation. The histogram of the “ d -only” Hubbardmodel corresponds to a single contribution near zero-order. For the t dd -PAM the low-expansion order featuregets instead less pronounced and, interestingly, a secondstructure develops at larger expansion orders. How canwe understand this new higher-order peak and can weassign a “label” to it?A first hint that the second peak at higher expansionorders reflects the presence of the p -orbital comes fromthe t pd dependence of its position. The expansion or-der histograms are very well fitted by a double gaussian function. The mean value of the high-order feature scalesquadratically with t pd (see inset of Fig.5).A more quantitative label is however needed. Thisis obtained by looking at the orbital, spin and expan-sion order resolved site-reduced density matrix which canbe directly measured within the CTQMC calculation.The density matrix itself, whose diagonals are the stateweights , is defined as ˆ ρ α = | α (cid:105)(cid:104) α | where | α (cid:105) is anatomic many-body state of the local part of the impu-rity Hamiltonian. In the present, simple, case of density-density interaction we have | α (cid:105) = | (cid:105) , |↑(cid:105) , |↓(cid:105) , |↑↓(cid:105) andonly the state weights are non-zero. The density-matrixˆ ρ α was previously used in similar contexts, e.g. Ref. 38,to obtain information about how much time the systemspends in a given local state. By resolving its measure-ment also in the expansion order, this quantity tells usthe probability to find the system in a certain atomicstate when there are a specific number of hybridizationevents with a given spin and orbital state.With this piece of information we can assign anexpansion-order dependent intensity to each histogram ofFig.5. In other words we look at the expansion order fora certain spin and orbital and show as color intensity thevalue of the state weight for a fixed atomic state at eachexpansion order. In Fig.6 we show this quantity for thecase of t pd = 0 .
75. This unveils a very interesting prop-erty of the hybridization-expansion CTQMC histogramfor the t dd -PAM with finite hybridization.In the Hubbard model case the order-resolved densitymatrix does not display a particularly strong expansion-order dependence. Therefore plotting the histogram withthe colors from the density matrix would not be partic-ularly revealing. Instead, in the case of the t dd -PAM thetwo peaks in the histogram are characterized by almostcompletely separated classes of diagrams, as indicatedby the complementary color intensities in panels ( ii ) and( iii ) of Fig.6. The density matrix used for the colors ofFig.6 is calculated for a fixed number k of pairs of oper-ators with spin σ = ↑ in the local trace. Panel ( iii ) show-ing ˆ ρ ↓ therefore tells us that a large number of diagramswith many ( i.e. high expansion-order) spin- ↑ electronshopping from and to the impurity contribute to a mea-sure of the local state |↓(cid:105) . This means that the impurityoften visits the state |↑↓(cid:105) , i.e. the hybridization with thebath makes it often doubly occupied. Since t pd is largeand the p -band is almost filled ( (cid:104) n p (cid:105) (cid:39) p -orbital acts as a very efficient particle-donor with re-spect to the impurity, indicating that the peak at large k describes hybridization processes of mostly p -character.We have checked that in a specular situation, with the p -orbital almost empty, the peak at large k has intensecolor for ρ ↑ , i.e. it corresponds to diagrams “emptying”the impurity. Since in that case the p -orbital “accepts”electrons the same conclusion of the large- k peak beingmostly of p -character holds.The interpretation of the large- k peak as stemmingmainly from the hybridization with the p -orbital suggeststhe following consideration. Since in the hybridization-expansion CTQMC the mean value of the expansion or-der histogram is proportional to the kinetic energy , for t pd (cid:54) = 0 we can identify the presence of two distinct com-ponents in the system, one with smaller kinetic energypredominantly of d -character and a more mobile one of p -character (see Figs.5 and 6). The latter componentis characterized by a large expansion order therefore itcorresponds to large hybridization strength.Let us note that we cannot directly relate peaks in theexpansion-order histogram to specific frequency struc-tures of χ spin ( ω ). Nevertheless, the previous analysis ofthe expansion-order resolved density-matrix allowed usto indirectly relate the presence of the feature at large k ,containing contributions to the impurity screening com-ing from mainly p -electrons, to the d - p character of thespin susceptibility. A more formal connection betweenthe expansion-order histogram and response functionsof the impurity model can be established by evaluatinghigher moments of the distribution. For example thewidth of the second peak in h ( k ) can give informationabout “ p -only” contributions to impurity susceptibilities.The present study provides a basis for such an analysis,which we leave for a future investigation. V. CONCLUSIONS
In this paper, using DMFT we have investigated a sim-plified, yet generic, model for d - (or f -) orbital materi- als, explicitly including hybridization with more itiner-ant, e.g. ligand p -, orbitals. We focused on the paradig-matic example of doped charge-transfer insulators. Inparticular, we studied the evolution of the dynamical spinresponse χ spin ( ω ) as a function of the hybridization. Wepointed out the existence of different exchange mecha-nisms involved in the local moment screening. We showedthat the direct exchange between d -orbitals, which char-acterize the screening physics of Hubbard-like models,competes with indirect exchange mechanisms (Kondosinglet formation) involving hybridization with conduc-tion band electrons. We show how the presence of suchdifferent exchange mechanisms is reflected in the struc-ture of the spin susceptibility. The low-frequency featureassociated to the metallic screening of local moments ofthe Hubbard model, acquires a multi-peaked structurescontaining contributions from both direct and indirectprocesses in the hybridized system. Moreover, the pres-ence of additional screening channels is mirrored in thedynamical spin response by the formation of spectralweight at intermediate energies, extending over an energyrange of the order of the conduction electrons bandwidth.Using CTQMC, we characterized the different processesinvolved in the dynamical screening of instantaneous lo-cal moments in terms of diagrams in the hybridization-expansion around the atomic limit. We show that inthe presence of finite hybridization, the expansion-orderhistogram acquires a characteristic double-peak struc-ture revealing the concomitant presence of a more lo-calized and a more mobile electronic component. A spe-cial analysis of the expansion-order state-resolved den-sity matrix allows us to assign a meaning to the peaksin the CTQMC histogram, associating them to d - or p -hybridization events separately. Our approach can bevery useful in realistic calculations for quantifying therelative importance and the degree of intertwinement ofthe different screening channels of local moments in mate-rials with orbitals of different degree of localization such, e.g. metallic cobaltates . Acknowledgments.
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