Characteristic Timescales of the Local Moment Dynamics in Hund's-metals
C. Watzenböck, M. Edelmann, D. Springer, G. Sangiovanni, A. Toschi
CCharacteristic Timescales of the Local Moment Dynamics in Hund’s-metals
C. Watzenb¨ock a , M. Edelmann b , D. Springer a , G. Sangiovanni b , and A. Toschi a a Institute of Solid State Physics, TU Wien, 1040 Vienna, Austria and b Institut f¨ur Theoretische Physik und Astrophysik and W¨urzburg-Dresden Cluster of Excellence ct.qmat,Universit¨at W¨urzburg, 97074 W¨urzburg, Germany (Dated: August 10, 2020)We study the characteristic timescales of the fluctuating local moments in Hund’s metal systemsfor different degrees of correlation. By analyzing the dynamical spin susceptibility in the real-timedomain we determine the timescales controlling oscillation and damping of on-site fluctuations - acrucial factor for the detection of local moments with different experimental probes. We apply thisprocedure to different families of iron pnictides/chalcogenides, explaining the material trend in thediscrepancies reported between experimental and theoretical estimates of their magnetic moments.
PACS numbers: 71.27.+a, 75.20.Hr, 71.10.Fd
Introduction. – Our perception of the natural world issignificantly shaped by the properties of the detectionprocess considered. One crucial aspect is the timescale ofthe probing mechanism: If this is larger than the typicaltimescale of the phenomenon under investigation, onlyaveraged information will be gained. This general state-ment applies to a very broad class of detectors, rang-ing, e.g. from the vision process in our eyes to the caseof interest for this work: the measurement of magneticproperties in correlated materials.Here, we focus on the detection of the local magneticmoments in correlated metallic systems. Their properdescription is, indeed, a key to understanding many-electron systems beyond the conventional band-theoryframework, being central to: Kondo physics[1, 2], Mott-Hubbard[3–5] or Hund-Mott[6–10] metal-insulator tran-sitions, quantum criticality of heavy fermion systems[11,12], magnetic and spectroscopic properties of Ni andFe[13–15] and of unconventional superconductors[16, 17].Reflecting the high physical interest, several exper-imental procedures are used to detect the local mag-netic moments and their manifestations[18]: measure-ments of static susceptibilities [13, 18], inelastic neutronspectroscopy (INS) [19], by integrating over the Brillouinzone(BZ) [20], x-ray absorption or emission spectroscopy(XAS or XES), etc.Whether it is possible to obtain an accurate descrip-tion of the local moments largely depends on the re-lation between the intrinsic timescales of the exper-imental probes and those characterizing the dynami-cal screening mechanisms at work. The emerging pic-ture is typically clear-cut if the screening processes arestrongly suppressed: In Mott or Hund’s-Mott insulat-ing phases coherent description of the magnetic momentproperties can be easily obtained in all experimental se-tups. A more complex, multifaceted situation charac-terizes systems where well preformed magnetic momentspresent a rich dynamics. Good examples are the stronglycorrelated metallic regimes adjacent to a Mott metal-insulator transition, or even better, compounds display- ing a Hund’s metal behavior[6, 21], such as iron pnictidesand chalcogenides[17].In this work, we illustrate how to quantitatively es-timate the characteristic timescales of fluctuating mo-ments in many-electron systems within the regime of lin-ear response. As a pertinent example, we apply thisprocedure to investigate the puzzling discrepancies be-tween experimental and theoretical estimates of the mag-netic moment size in the different families of iron pnic-tides/chalcogenides, clarifying the peculiar material de-pendence of this long-standing issue.
An intuitive picture. – For a transparent interpretationof our realistic calculations, we start from some heuris-tic considerations on the dynamics of the local magneticmoment ~µ = g µ B ¯ h ~S in a correlated metal. The relevantinformation is encoded in the time dependence of its cor-relation function F ( t ) ≡ g µ B ¯ h h{ ˆ S z ( t ) , ˆ S z (0) }i , (1)where g ∼ = 2 is the Land´e factor, µ B the Bohr magne-ton and ˆ S z = P ‘ ˆ s ‘z the z -component of the total spinmoment hosted by the correlated atom (e.g., a transitionmetal element), built up by the unpaired electronic spins s z of its partially filled d or f shells [18]. We stress thatEq. (1) describes both the static (thermal) and dynamic(Kubo) part of the response [22], which is needed forour study. In general, one expects the maximum valuesof F ( t ) at t = 0: This describes the instantaneous spinconfiguration of the system, often quite large in a mul-tiorbital open shell due to the Hund’s rule. Because ofelectronic fluctuations, the probability of finding a mag-netic moment of the same size and the same orientationwill be decreasing with time. At a first approximation,one can identify two distinct patterns for this process:(i) a gradual rotation (with constant amplitude) and (ii)a progressive reduction of the size of the local moment.Within this simple picture, two characteristic time (andenergy) scales for the local moment dynamics are natu-rally defined: (i) the period of the rotation ( t ¯ ω ∝ ω ) and(ii) the characteristic time ( t γ ∝ ¯ hγ ) for the amplitude a r X i v : . [ c ond - m a t . s t r- e l ] A ug FIG. 1. Schematic representation of the time decay of localspin correlations in the underdamped/overdamped regimes. damping.The values of the characteristic timescales may varyconsiderably from one material to another, with overalllarger values associated to a suppressed electronic mobil-ity. In the extreme case of a Mott insulator, one expectsto observe long-living magnetic moments, consistent withthe analytic divergence of the timescales found in thefully localized (atomic) limit ( t ¯ ω , t γ → ∞ ). On the op-posite side, in a conventional (weakly correlated) metalboth scales will be extremely short, roughly of the or-der of the inverse of the bandwidth W of the conductingelectrons ( t ¯ ω ∼ t γ ∝ ¯ hW ). The most interesting situation isrealized in a correlated metallic context. Here, the slow-ing down of the electronic motion, induced by the elec-tronic scattering, increases the values of both timescalesthat remain finite, nonetheless. The enhancement willdepend on specific aspects of the many-electron prob-lem considered, possibly affecting the two timescales ina different fashion: This leads to the distinct regimes of underdamped ( t γ (cid:29) t ¯ ω ) and overdamped ( t γ (cid:28) t ¯ ω ) localmoment fluctuations, schematically depicted in Fig. 1.The actual hierarchy of the timescales will strongly im-pact the outcome of spectroscopic experiments. Further,quantitative information about the dynamics of the mag-netic fluctuations at equilibrium may also provide im-portant information for the applicability of the adiabaticspin dynamics[23–25] and, on a broader perspective, cru-cial insights for the highly nontrivial interpretation of the out-of-equilibrium spectroscopies. Quantification of timescales. – The procedure toquantitatively estimate the characteristic timescales frommany-electron calculations and/or experimental mea-surements relies on the Kubo-Nakano formalism for linearresponse. Here, we recall that the dynamical susceptibil-ity is defined as χ ( τ ) ≡ h T τ ˆ S z ( τ ) ˆ S z (0) i (2)in imaginary time ( T τ is the imaginary time-ordering op-erator). The corresponding (retarded) spectral functions χ R ( ω ) are obtained via analytic continuation of Eq. (2).The absorption component of the spectra, Im χ R ( ω ), di-rectly measurable (e.g. in INS), provides a direct route for quantifying the timescales. In particular, simple an-alytic expressions, directly derived for damped harmonicoscillators, can be exploited for fitting the (one or more)predominant absorption peak(s) of Im χ R ( ω ). In the il-lustrative case discussed above, one hasIm χ R ( ω ) = A γω ( ω − ω ) + 4 ω γ , (3)where γ and ω are the scales associated to the majorabsorption processes active in the system under consid-eration (with ¯ h = 1), and the constant A reflects the sizeof the instantaneous magnetic moment. The expressionis clearly generalizable to other cases, where more ab-sorption peaks are visible in the spectra, as a sum of thecorresponding contributions [26].The full time-dependence of the fluctuating local mo-ment, which will reflect the interplay of the timescalesdefined above, is eventually obtained via the fluctuation-dissipation theorem F ( t ) = π R ∞ d ω cos( ωt ) coth( β/ ω ) Im χ R ( ω ) , (4)where β = ( k B T ) − is the inverse temperature. The case of the Hund’s-metals. – While the procedureillustrated above is applicable to all spectroscopic ex-periments of condensed matter systems, we will demon-strate its advantages for studying Hund’s metals[6, 21],where the dynamics of fluctuating moments is of partic-ular interest [27]. These systems can be viewed as a new“crossover”-state of matter, triggered by sizable valuesof the local Hubbard repulsion ( U ) and Hund’s rule cou-pling ( J ), when the corresponding atomic shell is (about)one electron away from a half-filled multiorbital configu-ration. At strong coupling, the interplay between U and J can induce either a Mott or a charge-disproportionateHund’s insulator [9, 28]. Out of half-filling, the compe-tition between these two tendencies can also stabilize ametallic ground state in the presence of high values of theelectronic interaction[6, 9, 28, 29]. The emerging physicsof a large local magnetic moment fluctuating in a stronglycorrelated metallic surrounding evidently represents oneof the best playgrounds to applying our time-resolvedprocedure.The prototypical class of materials displaying Hund’smetal physics is represented by the iron pnictides orchalcogenides. These compounds, which often displayunconventional superconducting phases upon doping, arealso characterized by interesting magnetic properties[17, 20, 30]. Both the ordered magnetic moments (mea-sured by neutron diffraction in the magnetically orderedphase) and the fluctuating moments (measured by INSin the paramagnetic high- T phase) are reported to besystematically lower[31] in experiment than in (static) lo-cal spin density approximation (LSDA) calculations (pre-dicting a large ordered moment of about 2 µ B for almostall compounds of this class). It was also noted that, β /2 Bare Bubble DMFT Bubble Full Calculation χ ( τ ) [ µ B ] τ LaFeAsOBaFe As LiFeAsKFe As FeTe 0 0.5 1 1.5 2 2.5 0 5 10 15 20 β /2 Bare Bubble DMFT Bubble Full Calculation τ β /2 Bare Bubble DMFT Bubble Full Calculation τ Bare Bubble DMFT Bubble Full Calculation I m χ R ( ω ) [ µ B / e V ] ω [eV] Bare Bubble DMFT Bubble Full Calculation ω [eV] Bare Bubble DMFT Bubble Full Calculation ω [meV] -1.5-1-0.5 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 3 Bare Bubble DMFT Bubble Full Calculation F ( t ) [ µ B ] t [fs] -0.5 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 3 Bare Bubble DMFT Bubble Full Calculation t [fs] -1 0 1 2 3 4 5 6 7 0 10 20 30 40 50
Bare Bubble DMFT Bubble Full Calculation t [fs]
FIG. 2. Spin susceptibility of the 3 d -Fe atoms as a function of imaginary time (first row), corresponding absorption spectra inreal frequency (second row) and correlation function in real time (third row), computed for different families of iron pnictidesor chalcogenides at β = 50 eV − ( T ≈ surprisingly, the larger discrepancies are found for the“less correlated” families 1111 (e.g. LaFeAsO) and 122(e.g. BaAs O ), which display milder quasiparticle renor-malization effects and are characterized by lower valuesof the screened Coulomb interaction estimated in con-strained random phase approximation (cRPA)[32]. Sig-nificantly smaller (or almost no) deviations are reported,instead, for the most correlated families such as the 11subclass (e.g. FeTe), where relatively large local momentsare found both in neutron experiments and theory. Pre-vious dynamical mean-field theory (DMFT) studies ofthe INS results suggested[33–36] that the local spin fluc-tuations on the Fe atom–whose time-resolved descriptionis the central topic here–may be responsible for the ob-served discrepancies. These works were restricted to onecompound or (at most) one family only, and did not an-alyze the real-time domain. Hence, no definitive con-clusion could be drawn about this issue, motivating thepresent computational material study. Ab-initio + DMFT calculations. – We report here onour density functional theory (DFT) + DMFT calcula-tions [37, 38] of the local spin susceptibilities in the iron pnictides/chalcogenides. Different from preceding works,we computed the spin-spin response functions on equalfooting for several different compounds, chosen as repre-sentative of the most relevant families (1111, 122, 111,11). As a step forward in the theoretical description,we put emphasis on a quantitative time-resolved analysisof the results, eventually allowing for a precise interpre-tation of the physics at play and of the spectroscopicresults.For our DMFT calculations[26, 39], we considered aprojection on the Fe-3 d (maximally localized) Wannier-orbital manifold. We assume an on-site electro-static interaction with a generalized (orbital-dependent)Kanamori form. The corresponding Hamiltonian reads: H = X k σlm H lm ( k ) c † k lσ c k mσ + H int (5)where l, m are orbital indices, k denotes the fermionic ω [eV] γ [eV] t γ [fs] t ¯ ω [fs] t [fs]LaFeAsO As As ω and γ of the absorptionpeak(s) computed in DMFT with Eq. (3) (first and secondcolumn, where the largest energy scale is marked in bold);effective lifetime χ ( t → ∞ ) ∝ e − t/t γ (third column); effec-tive oscillation period t ¯ ω = ¯ h/ p ω − γ (fourth column) and t = h ¯ h/ Z i ImΣ i ( ω → i all orb. (fifth column) is the effec-tive orbital averaged one-particle lifetime for the different ma-terial considered. See [26] for further details. momentum, and σ, σ the spin, and H int = P r l U ll n r l ↑ n r l ↓ + P r σσ ,l 1) is the Bose-Einstein distri-bution function (with ¯ h = 1). This is especially rele-vant for the “less correlated” compounds (LaFeAsO andBaFe As ), where t γ , t ¯ ω < t INS . In families with higherdegrees of (e.g. for FeTe, where t γ , t ¯ ω > t INS ) the av-eraging effect gets “mitigated”, allowing the detection oflarger magnetic moment sizes, consistent with fast probeXAS and XES experiments[44, 45]. The material depen-dence of local moment dynamics is directly mirrored inthe progressive red shift of the first-absorption peak inIm χ R ( ω ), as shown in Fig. 3. Here, one can appreciatehow an increasing part of the spin absorption spectragradually enters the accessible energy window of the INS(main panel). This explains the progressively reduceddiscrepancies in the size of the magnetic moment (see in-set) observed in the more correlated families of the ironpnictides or chalcogenides. Conclusions. – We illustrated how to quantitatively in-vestigate, on the real-time domain, the dynamics of mag-netic moments in correlated systems and how to phys-ically interpret the obtained results in terms of theircharacteristic timescales. Our procedure, exploiting thefluctuation-dissipation theorem, is then applied to clarifythe results of INS experiments in several families of ironpnictides and chalcogenides. In particular, the differentdegrees of discrepancies with respect to the standard abinitio calculations is rigorously explained by comparingthe timescales of the fluctuating moments to the char-acteristic timescale of the INS probe. Remarkably, thestrong differentiation among the timescales of the ma-terials considered, crucial for a correct understanding ofthe underlying physics, is almost entirely due to vertexcorrections.While the dynamics of the magnetic moments isparticularly intriguing in the Hund’s metal materialsconsidered here, the same procedure is directly appli-cable to all many-electron systems and to fluctuationsof different kinds[2]. A precise quantification of thecharacteristic timescales may provide new keys to con-nect the findings of equilibrium and out-of-equilibriumspectroscopies, as well as crucial information on theapplicability of adiabatic spin dynamics approaches[25]. Acknowledgments. We thank B. Andersen, L. Boeri,M. Capone, L. de’ Medici, P. Hansmann, K. Held,J. Tomczak and M. 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Toschi a 1, 2 a Institute of Solid State Physics, TU Wien, 1040 Vienna, Austria b Institut f¨ur Theoretische Physik und Astrophysik, Universit¨at W¨urzburg,Am Hubland Campus S¨ud, 97074 W¨urzburg, Germany COMPUTATIONAL DETAILS Density Functional Theory - For the density functional theory (DFT) calculations we employed the VASPcode[1, 2], version 5.3.3. As structural inputs, the experimentally found crystal structures as well as the measuredlattice parameters (given in table I) have been used.For all of the atoms in the given structures, we used PBE-GGA functionals. The precise functionals used for eachatom are given in table II. Calculations were performed on a Γ-centered MP k-mesh with 12 × × 12 points and10 × × 12 points for I4/mmm and P4/nmm structures, respectively; The partial occupancies were calculated usingthe Bl¨ochl tetrahedron method. The respective cut-off energies were, among other parameters, defined by setting theprecision to HIGH, and the DOS was evaluated on 2001 points. Material Crystal structure Space group a [˚A] c [˚A] zLaFeAsO ZrCuSiAs-type P4/nmm 4.0355[3] 8.7393[3] 0.1418 La , 0.6507 As [4]LiFeAs PbFCl-type tetragonal P4/nmm 3.774[5] 6.354[5] 0.8459 Li , 0.2635 As [6]BaFe As ThCr Si -type I4/mmm 3.9625[7] 13.0168[7] 0.3545 As [7]KFe As ThCr Si -type I4/mmm 3.842[8] 13.861[8] 0.3525 As [8]FeTe PbO-type P4/nmm 3.8279[9] 6.2561[9] 0.285 Te [10]TABLE I: Crystal structures for all materials under consideration. For I4/mmm materials, c is given as the lattice parameterof the tetragonal cell, and z in relation to this c .Element Creation date VHRFINLa Sep 6th 2000 core Kr4dFe Sep 6th 2000 d7s1Ba Sep 6th 2000 5s5p6sTe Apr 8th 2002 s2p4O Apr 8th 2002 s2p4K Jan 17th 2003 p6s1Li Jan 17th 2003 s1p0As Sep 22nd 2009 s2p3TABLE II: List of PAW PBE functionals used by VASP in the DFT calculations of this study. The functionals are uniquelyidentified by their creation date and the valence electron configuration given in the functionals by VHRFIN. Wannier Projection - The VASP results were projected onto local orbitals via the wannier90 code[11]. At the timeof the calculations, wannier90 integration into VAPS was only possible with wannier90 v1.2. Specifically, all of theelectronic Bloch functions in the DFT calculation were projected onto the d states of Fe, no bands were marked asexcluded in the wannier90.win file. In wannier90, the k-points were identical to those from the respective VASPMP-grids. The electronic bands of predominant Fe character are intertwined with bands of other character, such asthe p states of the ligands. The degree of entanglement varied across materials, necessitating different disentanglementwindow parameters for wannier90, and in the case of BaFe As and KFe As also frozen windows. The window a r X i v : . [ c ond - m a t . s t r- e l ] A ug positions were tweaked manually with respect to disentanglement convergence as well as agreement between originalVASP bands and the bands of the Wannier Hamiltonian, the final values are given in table III. The convergencecriterion for the disentanglement as well as the wannierization was a difference in spread between successive iterationslower than 10 − . Best results were achieved by enabling guiding centres. The Wannier Hamiltonian served as thesingle-particle Hamiltonian for the Dynamical Mean Field Theory (DMFT) calculations. Material Disentangle window [eV] Frozen window [eV]min max min maxLaFeAsO -2.0473 2.4527LiFeAs -2.4441 2.7559BaFe As -1.0723 2.7277 -1.0723 -0.3723KFe As -4.7175 3.2825 -1.3175 3.0825FeTe -2.0831 2.4169TABLE III: Energy windows for disentangling of bands in wannier90. KFe As and BaFe As additionally required windowsdefining frozen states. Energies are given relative to the Fermi energy E F =0 eV. Dynamical Mean Field Theory - To include the effects of strong local interactions on top of the DFT, we performedDMFT simulations of an low-energy model for the entire 3 d -orbital manifold of Fe. The most general form of anon-site electrostatic repulsion in this manifold reads H int = X r σσ X lmno U lmno c † r lσ c † r mσ c r oσ c r nσ , (1)where the full-fledged, four-indexed U − tensor describes the projected value of the screened Coulomb interaction onthe corresponding orbital configurations. As an ab-initio estimate for the orbital-dependent interaction parameters,we take the results by Miyake et al.[12], where constrained random phase approximation (cRPA) results for the two-orbital interaction matrix U lm and J lm were reported for different compounds[26]. Here, the J lm values encode the(orbital-dependent) Hund’s coupling, while the U ij diagonal/off-diagonal matrix elements describe the inter-/intra-orbital electrostatic repulsion. The relation, which we exploited to extract the interaction parameters appearing inEq. 1, is: U ijkl = U ij , if ijkl = ijij,J ij , if ijkl = iijj and i = j,J ij , if ijkl = ijji and i = j, , otherwise . . (2)This leads to the low-energy Hamilonian used for our DMFT calculations H = X k σlm H lm ( k ) c † k lσ c k mσ + H int , (3)with H int = P r l U ll n r l ↑ n r l ↓ + P r σσ P l LaFeAsO 2.53 0.39BaFe As As d − model (in eV). robustness of our conclusions, we have also performed DMFT calculations using the orbitally-averaged values for the U and J interaction (i.e., corresponding to a “conventional” Kanamori interaction, not shown), finding only marginalchanges to the results shown in Fig. 2 and 3 of the main text. Larger quantitative modifications can be found in theresults of the most correlated materials, as expected, only if one neglects the spin-flip terms in Eq. (5) (e.g., whenperforming density-density calculations, not shown here[13]). The reason is, that in this approximation one tends tooverestimate the high-spin configurations in the strong-coupling regime.The number of electrons in the target ( d -) manifold was estimated directly from chemical considerations (constituentelectronegativity). Throughout our calculation, we assumed that LaFeAsO, BaFe As LiFeAs and FeTe have a fillingof P l,σ h n lσ i = 6 . As we used, instead, P l,σ h n lσ i = 5 . µ F LLDC ( i ) = µ F LLDC ( i ) + (cid:0) n − (cid:1) (cid:16)P j ( U ij − J ij ) (cid:17) , (6)In eq. (6) n = l +1) P i,σ n i,σ is the DFT filling and the two-indices U-matrix is related to the four-indices local(screened) Coulomb-tensor by U ij = U ijij and J ij = U ijji (with i = j ). LaFeAsO BaFe As LiFeAs KFe As FeTe3 z − r xz yz x − y xy The DMFT simulation was performed with a continuous-time quantum Monte Carlo (QMC) algorithm implementedin the code package w2dynamics [16]. All calculations were done at β = 50[eV − ] corresponding to approximately232 . ÷ Nmeas =10 , where Nmeas is then number of QMC measurements, see [16] for details).Up to 100 additional DMFT-steps were performed with higher statistics.For each calculations, the final convergence of the DMFT self-consistency was tested for the one-particle quantitiesencoded in the self-energy Σ l ( ω ), with respect to the previous five iterations. For the number of steps betweenmeasurements (where the minimum value gives a measure of auto-correlation time) we found a value of Ncorr =1500 ÷ Nmeas > · . As aQMC-sampling algorithm we applied the recently developed States-Sampling [20]. The result is show in the upper-right part of Figure 2 of the main text.To obtain the DMFT-bubble susceptibility we used χ Bubble − DMFT ( τ ) = P l G l loc ( τ ) G l loc ( β − τ ), and for the bare-bubble we set the interaction as well as the double counting correction in the DMFT calculation to zero ( U = J = V = µ DC = 0), corresponding to G loc = G . Analytical continuation - Analytical continuation from imaginary time (where the QMC data was obtained) to realfrequencies was performed with the Maximum Entropy Method (MaxEnt) with the code package Maxent [21]. Thisway we obtained the imaginary part of the retarded susceptibility χ R ( t ) ≡ i ~ θ ( t ) h [ ˆ S z ( t ) , ˆ S z (0)] i . The effect of differentdefault-models (Flat, Gaussian, Lorentzian) was tested and found to be small. We chose a broad featureless Lorentzian-default model with a width of γ Model = 0 . 5. Model-details are found elsewhere [21]. The optimal α − parameter (weightof the entropy term in MaxEnt) was determined by the maximum of the curvature of χ ( α ). (See fig. 1.) This waywe could reliably determine the region where neither the data was over-fitted nor the default model was take intoaccount too strongly. One advantage of this method is invariance of the final spectrum under re-scaling the error bya global factor[13]. A similar approach was already applied in [22]. f ( ) ( ) 10 f () / () FIG. 1: Log-log-plot of the quadratic difference between the data and the fit χ over the entropy parameter α for LaFeAsO.We find overfitting (underfitting) of the data to start at α < ( α > ). The spectrum corresponding to the maximum of f ( α ) /χ ( α ) (at α = 2 · ) could be regarded as a good analytical continuation. One particle time scales - For estimating one particle timescales we assumed that (in the presence of a well de-fined quasi-particle excitation) the one-particle Green’s function G i ( t ) for each orbital i decays in the followingway: (cid:12)(cid:12) G i ( t ) (cid:12)(cid:12) = Z i e − t Zi ImΣ i ( ω → ~ ∝ e − tti , with t i ≡ ~ Z i ImΣ i ( ω → , where Σ i is the self-energy of the orbital i .The value of the self-energy at zero frequency as well as the orbital dependent quasi-particle mass re-normalization Z i = (cid:0) / d ω ReΣ i ( ω ) (cid:12)(cid:12) ω → (cid:1) − was extracted from the DMFT self-energy by linear interpolation of ImΣ(i ω n → Z i ). The one particle time scale t given in the main text was thenestimated as the orbital average of t i : t = P i t i . Spin-excitation time scales - While the time scales of spin-excitations in iron-based superconductors are determinedby an intricate interplay of kinetic energy (hopping) and electron-electron-interaction the main time scales can beeffectively described by a much simpler model. The extraction of time scales was done by applying a uniform χ -fitto Im χ R ( ω ) with cutoff-values chosen for the grid such that the main-peaks structure is well within the frequencywindow (1eV). The cutoff excludes high-frequency data, which is usually not as well captured by MaxEnt as thelow-frequency data. A variation of the cutoff by 20% leads to a change in the time scales by less than 15%.The fitting function is defined as follows: We consider the absorption spectrum of a damped harmonic oscillator,which can be obtained by the Fourier-transform of the Green’s function of the differential equation ¨ χ ( t ) + 2 γ ˙ χ ( t ) − ω χ ( t ) = − δ ( t ), i.e. χ ( ω ) = ω − γω + ω . We note that the latter has poles only on the lower half-plane, andthus it is a retarded function ( χ ( t < 0) = 0). Its imaginary part (up to a proportionality-constant reflecting thematerial-dependent value of the unscreened local moment) defines our fitting model which, thus, readsIm χ R ( ω ) = 2 γω ω − ω ) + 4 ω γ , (7)or correspondingly in real times χ ( t ) = e − γt √ ω − γ sin( p ω − γ t ) θ ( t ) if ω > γ − γt √ γ − ω sinh( p γ − ω t ) θ ( t ) if ω < γ . (8)The asymptotic behavior, which determines the main-lifetime is given bylim t →∞ χ ( t ) ∝ ( e − γt ≡ e − t/t under γ if ω > γ e − (cid:16) γ − √ γ − ω (cid:17) t ≡ e − t/t over γ if ω < γ . (9)The corresponding parameters obtained by fitting the DMFT spectra are summarized in table VI. ω [eV] γ [eV] t γ [fs] t ¯ ω [fs]LaFeAsO As As χ ( t →∞ ) ∝ e − t/t γ (third column) and effective oscillation frequency t ¯ ω = ~ √ ω − γ (fourth column) One can also define a harmonic-oscillator anti-commutator through the fluctuation-dissipation theorem as F ( ω ) = π coth( ωβ/ χ R ( ω ). For the latter it is not easy to get an analytical expression for the Fourier-transform( F harm . osz . ( t ) = R ∞−∞ d ω e − i ωt π coth( ωβ/ χ data ( t ) against the analytical expressions given in eq. (8). The results of thetransformation of the data as well as the transformation of the fits is shown in fig. 2. For LaFeAsO, BeFe As ,LiFeAs and KFe As a single peak model was used, while for FeTe the double-peak-structure in the data necessitateda two-peak model. Due to the second peak in FeTe no sign-change is observed in the corresponding χ ( t ), althoughthe main (first) peak would predict an oscillatory (under-damped) behavior. I m R () LaFeAsO dataLaFeAsO fitBaFe As dataBaFe As fitLaFeAs dataLaFeAs fitKFe As dataKFe As fitFeTe dataFeTe fit 0 10 20 30 40 50t [fs]0.000.250.500.751.001.251.501.75 ( t ) data LaFeAsOdata BaFe As data LaFeAsdata KFe As data FeTe model LaFeAsOmodel BaFe As model LaFeAsmodel KFe As model FeTe 0 10 20 30 40 50t [fs]10123456 F ( t ) data LaFeAsOdata BaFe As data LaFeAsdata KFe As data FeTe trans. fit LaFeAsOtrans. fit BaFe As trans. fit LaFeAstrans. fit KFe As trans. fit FeTe FIG. 2: Dissipative part of the spin-spin susceptibility obtained by MaxEnt (left figure solid lines). The harmonic-oscillatorfits are shown as dashed lines. Center figure: Spin-spin susceptibility in time. Direct transform of MaxEnt data shown assolid lines and the analytic expression for the fitted model as dashed lines. 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