Magnon damping as a probe of Kondo coupling in magnetically ordered systems

Domain: chemistry
Published: 2026-03-06
Editorial outcome: major_revision

Abstract
In d-electron systems, there can also be intricate interplay between Kondo coupling and magnetic interactions as that in f-electron systems, but the underlying mechanism remains elusive. Here, using inelastic neutron scattering, we investigate the temperature evolution of the low-energy spin waves (magnons) in a metallic van der Waals ferromagnet Fe3−xGeTe2, and observe that the magnon damping diverges at both low and high temperatures and exhibits a minimum at an intermediate temperature. These behaviours are described by a formula that combines logarithmic and power-law terms, representing the dominant contributions from Kondo coupling and thermal fluctuations, respectively. These findings can be explained by considering electron-magnon scattering of spin-flip type within the ferromagnetic Kondo-Heisenberg lattice model, distinct from the original Kondo effect which only considers the coupling between itinerant electrons and isolated impurity spins. These results unveil the intriguing interplay between itinerant electrons and spin waves in metallic 3d-electron systems with magnetic order, and provide magnon damping as a new effective probe of Kondo coupling in metallic quantum magnets, thereby opening new avenues for exploring Kondo physics from the magnon perspective.

Introduction
The original Kondo effect in dilute magnetic alloys, which takes the resistivity minimum as one of the characteristics, describes the scattering of conduction electrons by magnetic impurities in such systems [1, 2]. This effect is classified as the single-impurity Kondo problem and is now well understood based on the spin-flip scatterings of conduction electrons off the single-impurity spins (Fig. 1a) [1, 2]. The concept of Kondo physics has been extended to include the correlated electron systems, specifically those consisting of a dense periodic array of local moments interacting with the conduction electron sea through an antiferromagnetic Kondo interaction, referred to as a Kondo lattice (Fig. 1b-d) [2–5]. The Kondo-lattice model was initially proposed to describe the heavy-fermion state in f-electron systems, particularly in intermetallics containing Ce, Yb, or U elements [2–7]. The competition between the Kondo effect and couplings of localized spins underlies various intriguing phenomena in these heavy-fermion compounds (Fig. 1e), including quantum criticality [8–11], strange-metal behaviour [11, 12], magnetism [13–16] and superconductivity[17, 18], which can be tuned by chemical doping, magnetic field or pressure [8–12]. In the strong Kondo coupling regime, the hybridization between localized f electrons and conduction electrons leads to the quenching of local moments and the expansion of the Fermi surface, resulting in the formation of a heavy Fermi-liquid state (Fig. 1c,d) [2–7, 19, 20]. Magnetic order can arise when the strength of the effective magnetic interaction between localized spins exceeds the Kondo coupling. Moreover, the coexistence of magnetic order with Kondo effect [13–16] or superconductivity [17, 18] can be observed, particularly in systems where the duality of f electrons emerges due to the mixed-valence situation and the multiorbital nature [13–18].

More recently, the unexpected heavy-fermion state has also been observed in certain d-electron transition metals, which reside in the intermediate regime between itineracy and localization of d electrons [21–31]. Understanding magnetism within the framework of this duality and the applicability of the Kondo-lattice model to such systems are currently active topics of research [21, 23, 25–31]. The van der Waals (vdW) metallic ferromagnet Fe3−xGeTe2 is one such example material [29–35]. Due to the coexistence of localized and itinerant 3d electrons, there are conflicting reports regarding the microscopic origin of the magnetism in Fe3−xGeTe2 [29–32, 36–39]. In our earlier work, we have reconciled the debate by showing that the ferromagnetism has a dual origin, with local moments and itinerant electrons contributing to the low-energy spin waves and columnlike continua, respectively [31]. Moreover, there is accumulating evidence for the Kondo coupling between these two components, such as the Fano resonance feature [29, 30], large effective electron mass [29, 34, 37], and the incoherent-coherent crossover in transport and magnetic measurements [29]. In particular, the resistivity curve of Fe3−xGeTe2 reveals a slope change at a characteristic temperature of approximately 90 K, as depicted in Fig. 1d. Below this temperature, the Kondo-lattice behaviour emerges in the magnetic ordering phase, accompanied by the enhancement of Fermi surface volume and effective electron mass [29]. Notably, this resistivity behaviour aligns with observations in other d-electron systems exhibiting a heavy-fermion state [22, 24, 25], but is distinct from either the Kondo-impurity model or the Kondo lattice in f-electron systems (Fig. 1d). Since our earlier work has indicated that Kondo screening significantly enhances spin-wave damping at 4 K compared to 100 K [31], a question naturally arises: will the damping follow certain scaling behaviour due to the Kondo coupling effect?

Here we report inelastic neutron scattering (INS) measurements on the temperature evolution of low-energy spin waves in Fe-deficient Fe3−xGeTe2 single crystals with Curie temperature TC ∼ 160 K. Upon cooling from TC, we observe that the magnon damping rate initially decreases, reaching a minimum around T∗d ∼ 90 K, before rapidly increasing again. This behaviour follows a scaling relation combining both logarithmic and power-law terms. Additionally, we find that the in-plane Kondo coupling is stronger than the out-of-plane coupling, resulting in a softening of in-plane magnons at low temperatures. Using the state-of-the-art tensor-network algorithm [40–45], we reproduce the observed magnon damping minimum and logarithmic scaling within a ferromagnetic Kondo-Heisenberg (FMKH) model. Although the form is analogous to the original Kondo effect describing the coupling between itinerant electrons and impurity spins [1, 2], the underlying mechanism is essentially different — here, itinerant d electrons scatter the collective excitations of local moments via a spin-flip process induced by Kondo coupling. Our results provide compelling evidence for the existence of Kondo effect in the metallic ferromagnet Fe3−xGeTe2, and demonstrate magnon damping as an effective probe for the Kondo coupling in systems where magnetic order and Kondo effect coexist.

Results
Damping rate minimum and logarithmic scaling

Figure 2a,b shows the energy scans at various temperatures for the off-centered in-plane and out-of-plane positions, respectively. As no sizable spin gap was observed at the zone center (0,0,2) [31], we chose Q positions away from the zone center in order to track the temperature evolution of the inelastic magnon peak. Intriguingly, for (-0.05,-0.05,2) within the in-plane direction (Fig. 2a), no magnon peak is observed at 4 K. Instead, the magnon peak emerges at higher temperatures, with its center exhibiting a nonmonotonic change approaching TC. For (0,0,1.7) in the out-of-plane direction (Fig. 2b), the peak center consistently shifts to lower energies with increasing temperature. To eliminate the influence of the Bose statistics, we correct the raw data with the Bose population factor, and plot them in Fig. 2c,d. We fit these corrected results by convoluting the momentum- and energy-dependent instrumental resolution with a damped harmonic oscillator (DHO) formula, which is applicable to damped spin waves [31, 46–48]. The DHO formula has a form of χ′′(Q,E) ∝ γE E0/[(E2−E02)2+(γE)2], where E0 is the magnon energy and γ is the damping rate (the inverse of γ is proportional to the lifetime of the damped magnons) [31, 46–48]. Based on these DHO fittings in Fig. 2c,d, we can extract the intrinsic damping rate and magnon energy. We emphasize that E0 represents the fitted magnon energy, which does not necessarily coincide with the peak maximum in the measured spectra. This is particularly the case for strongly damped magnons; for example, at 4 K, the peak maximum appears at a much lower energy than E0 (Fig. 2c).

Figure 2e,f presents the extracted γ and E0 plotted as a function of temperature, showcasing the most intriguing results of this work. It is found that for both in-plane and out-of-plane directions, γ shows an upturn toward both 0 K and TC, causing a minimum around T∗d ∼ 90 K. While disorder or vacancies may affect local moments and make spin waves broader, such effect, if present, is more likely to be temperature independent, and therefore would not be responsible for the observed nonmonotonic temperature dependence. On the other hand, this phenomenon is reminiscent of the resistivity minimum caused by the Kondo effect in the original single-impurity Kondo model (Fig. 1d) [1–4], where thermodynamic and transport properties depend logarithmically on temperature as −ln(T) [1–4]. Inspired by this, we use a similar logarithmic term to describe the divergent behaviour toward 0 K. In the meantime, a power-law term describing the spin-wave damping in the hydrodynamic regime is also required to explain the divergence toward TC [49, 50]. Actually, these two effects should exist simultaneously over the entire temperature range below TC. Therefore, the general formula should consist of the linear combination of the two terms, which reads as,

γ(T) = AIn(T∗d/T)+B(1−T/TC)−ν. (1)

To fit the data, we fix T∗d = 90 K and TC = 160 K, and allow A,B,ν to be free. The fitting results well reproduce the unusual nonmonotonic temperature evolution of γ for both in-plane (Fig. 2e) and out-of-plane (Fig. 2f) directions. Specifically, we obtain Ain = 1.617±0.204, Bin = 0.834±0.100, and νin = 0.715±0.087 for the in-plane direction, and Aout = 0.356 ± 0.024, Bout = 0.406 ± 0.025, and νout = 0.490 ± 0.040 for the out-of-plane direction. The distinct parameter values for the in-plane and out-of-plane directions indicate the presence of magnetic anisotropy in this material.

The extracted magnon energy E0 shown in Fig. 2e,f exhibits distinct behaviours for the in- and out-of-plane directions. With increasing temperature, it is found that the out-of-plane E0 softens through a power law while approaching TC (Fig. 2f), a behaviour similar to the temperature evolution of the magnetization in Fe3−xGeTe2 [36, 51]. On the other hand, the in-plane E0 keeps almost constant across T∗d and slightly softens upon cooling to lower temperatures (Fig. 2e). We attribute these two distinct behaviours of E0 to different degrees of Kondo coupling along the two directions. Notably, the magnitude of γ for the in-plane direction is much larger than that for the out-of-plane direction, being approximately four times greater at 4 K. This anisotropy can be explained by the quasi-two-dimensional vdW structure of Fe3−xGeTe2, which promotes a more itinerant electron character in the a-b plane and results in stronger in-plane Kondo coupling. Consequently, the in-plane Kondo screening markedly reduces the intralayer exchange coupling between local moments, leading to the observed decrease in the in-plane E0 at low temperatures (Fig. 2e).

Momentum dependence of the scaling behaviour

To examine whether these behaviours can also be observed in other momenta, we plot the results of the low-energy spin-wave excitations and their damping rate as a function of momentum in Fig. 3. We focus on the excitations at 90 K, where the spin waves are most coherent with the least damping. The energy scans along [110] and [001] directions are plotted in Fig. 3a,d. As Q increases, the peak centers shift gradually to higher energies, accompanied by a broadening of the linewidth and a weakening of the scattering intensities, indicating the nature of damped spin waves. We perform DHO fittings to these scans to extract the momentum dependence of E0 (Fig. 3b,e) and γ (Fig. 3c,f) for the two directions. The dispersions are fitted using the quadratic relation E0 = ∆+DQ2, where ∆ and D are the spin gap and spin-wave stiffness, respectively. The fittings yield D of 54.3±0.8 meV ˚A2 for the [110] direction (Fig. 3b) and 71.2 ± 1.3 meV ˚A2 for the [001] direction (Fig. 3e) at 90 K. For γ, we found a quasi-quadratic dependence (∼ q2.2) in the in-plane direction (Fig. 3c) and a linear dependence in the out-of-plane direction (Fig. 3f).

The low-energy spin waves at other temperatures like 4, 45 and 130 K are also measured. With increasing temperature, we find that the spin-wave dispersion remains nearly unchanged for the in-plane direction but softens considerably for the out-of-plane direction (Fig. 3b,e). Note that since the in-plane spin waves are heavily damped at 4 K, it is challenging to extract a reasonable dispersion. We find that the original q-dependences of γ at 90 K remain applicable at other temperatures, i.e., γ ∼ ηq2.2 and γ ∼ ηq, although the magnitude and coefficient η of the damping vary due to the temperature dependence of the Kondo screening effect (Fig. 3c,f). To further illustrate the temperature evolution of the spin waves, we plot the extracted spin-wave stiffness D and damping coefficient η as a function of temperature in Fig. 3g,h. The out-of-plane D follows a typical power-law relation with temperature, while the in-plane one remains nearly constant. For η, it follows a nonmonotonic dependence and can still be elucidated using Eq. 1 with fixed T∗d = 90 K and TC = 160 K for both directions. Together with Fig. 2e,f, these findings uncover a universal law independent of the momentum governed by the Kondo physics in Fe3−xGeTe2.

FMKH model and spin-flip scattering mechanism

Notably, the temperature at which this damping minimum occurs (Fig. 2e,f and Fig. 3h) is close to the incoherent-coherent crossover observed in the resistivity curve (Fig. 1d). Below the intermediate temperature, the localized electronic state hybridizes with the itinerant electronic state, enlarging the Fermi surface volume and effective electron mass[29]. Consequently, from the perspective of conduction electrons, the slope of resistivity changes (Fig. 1d), consistent with other experimental observations [29–31, 34], manifesting heavy-fermion behaviour. On the other hand, from the perspective of spin waves, these processes provide additional decay channels for magnons, resulting in a significant damping of magnons at low temperatures.

To understand the anomalous magnon damping behaviour microscopically, we develop an FMKH model, considering the dominant role of direct ferromagnetic (FM) exchange interactions between localized d moments and their competition with Kondo coupling in Fe3−xGeTe2 (Methods). The strong direct exchange interactions arise from the spatially extended nature of the outermost 3d orbitals, whose significant overlap stabilizes robust long-range FM order and a relatively high TC. This stands in sharp contrast to conventional f-electron Kondo-lattice systems, where the highly localized f orbitals are effectively screened by conduction electrons and the local f moments couple primarily via weaker, indirect Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions, which often favor antiferromagnetic order and yield lower ordering temperatures [8–10, 13, 14]. In Fe3−xGeTe2, the localized d moments are only partially screened by itinerant electrons (Fig. 1b). Within this framework, magnons as collective modes of local moments are interacting with conduction electrons via Kondo coupling [32, 52], giving rise to the Kondo effect [29–31]. To characterize such effect, we employ tensor-network methods to compute the dynamical properties of a minimal 1D FMKH system at finite temperature, represented by a two-leg ladder: one leg as a chain of local moments and the other as itinerant electrons (Fig. 4a) (see Methods for details).

The calculated damping rate for the FM phase of the FMKH model (JK/|JH|=1) is shown in Fig. 4c. A clear damping minimum at T∗d is observed, contrasting with the monotonic behaviour in the pure Heisenberg model (JK/|JH|=0). Notably, our simulations also reveal a logarithmic scaling γ(T) ∼ ln(T0/T) below T∗d. This damping minimum and logarithmic scaling behaviour can be observed in various momenta and for different Kondo couplings in the FM phase (Supplementary Fig. 2), indicating that it is a universal feature arising from the scattering between magnons and itinerant electrons.

We emphasize that the key mechanism underlying the anomalous magnon decay is spin-flip scattering between conduction electrons and magnons induced by the Kondo coupling. In these processes, a magnon decays into a fermionic particle-hole pair, providing an additional channel for damping (Fig. 4a). In the weak-coupling limit (JK ≪ |JH|) relevant to Fe3−xGeTe2 (Fig. 1e), the magnon self-energy is given diagrammatically in Fig. 4(b), where the renormalized vertex plays a critical role in enabling anomalous magnon decay [53]. In sharp contrast to the usual vertex correction due to electron-phonon coupling—which is negligible due to Migdal theorem [54] — here the vertex correction of JK in Fig. 4b is crucial at low temperatures. By applying the renormalization of the Kondo coupling, formally similar to the single-impurity Kondo problem but here used to describe the momentum-dependent renormalization and damping of magnons, a log(1/T)-like temperature dependence of the magnon self-energy can be obtained. Furthermore, as the kinematically allowed phase space for magnon-electron scattering increases with increasing magnon momentum Q (See Supplementary Fig. 4), the magnon damping becomes stronger with Q at low temperatures, in agreement with the experimental observations in Fig. 3.

Methods
Crystal growth and INS experiments. High-quality single crystals of Fe3−xGeTe2 were grown using self-flux method, as described in Refs. 31, 63. This growth method naturally introduces Fe vacancies, which reduce both the Curie temperature and the saturation magnetization. To make sure that different pieces of single crystals had the same amount of Fe vacancies, we measured the magnetization on four randomly picked single crystals from different growth batches. As shown in Supplementary Fig. 5, they have nearly identical susceptibility curves with a single TC ∼ 160 K, regardless the growth batch. This is in good agreement with our prior magnetization and resistivity measurements, as well as with the temperature evolution of the magnetic Bragg peak (Ref. 31). These results confirm the homogeneity of the crystals used in this study. The INS experiments were performed on Sika, a cold-neutron triple-axis spectrometer located at the OPAL facility of ANSTO in Australia [64, 65]. The Fe3−xGeTe2 sample array used in the measurements was the same as that employed in Ref. 31. It was composed of well coaligned single crystals with a mosaic of 1.6◦ within the a-b plane and 1.7◦ along the c axis (Supplementary Fig. 5), and mounted in the (H, H, L) scattering plane. Measurements were carried out using a fixed final-energy mode with Ef = 5.0 meV, where both incident and final neutron energies were determined by pyrolytic graphite (002) crystals. Double and vertical focusing modes were applied for the monochromator and analyzer, respectively. A cooled Be filter was placed after the sample during the first experimental run (corresponding to Fig. 2) and removed in the subsequent run, resulting in higher detected neutron intensity (corresponding to Fig. 3). An open-open-60′-60′ collimation was used to strike a fine balance between neutron flux and experimental resolution. These settings gave an energy resolution of ∼ 0.2 meV at the elastic line. To ensure sufficient statistics, each inelastic data point was counted for ∼ 5 minutes. Throughout the paper, the wavevector Q is expressed as (H, K, L) in reciprocal lattice unit (r.l.u.) of (a∗, b∗, c∗) = (4π/√3a, 4π/√3b, 2π/c) with refined lattice parameters a = b = 3.946 ˚A and c = 16.357 ˚A in a hexagonal structure.

FMKH model and tensor-network calculations. The FMKH model considered in this work reads

H = −t ∑n,σ (c†n+1,σ cn,σ +H.c.) + JH ∑n S n ·S n+1 + JK ∑n S n ·s n, (2)

where c†n,σ (cn,σ) is the creation (annihilation) operator of the conduction electron at site n with spin σ ∈ {↑,↓}, and S n (s n) represents the spin operator of local moment (conduction electron). The finite-temperature dynamical properties of this model are calculated by the tensor-network methods (for 1D chain) [40–45], and analyzed by perturbative calculations (also for higher dimensions [62]). The tensor-network calculation process can be divided into two parts: first, obtaining the equilibrium thermal density matrix, and second, performing real-time evolution on that density matrix. The equilibrium thermal density matrix and related thermodynamic properties can be obtained by the matrix product operator (MPO) based tensor-network methods [40–42]. Given the MPO representation of the thermal density matrix, we generalize the tangent-space approach for real-time evolution [43–45] to the mixed states and study the temperature evolution of the spectral density.

In the calculations, FM coupling strength JH = −1 is taken as the energy unit and the hopping energy is set as t/|JH| = 2 throughout the calculations. The simulations are restricted within the 2-leg ladder structure of size N = L×2 with the length L = 16. This moderate system size is chosen so as to guarantee converged dynamical data at finite temperature. One chain comprises local moments, while the other represents itinerant electrons. These two chains are coupled through Kondo interactions. The retained bond dimension is D = 1600, which guarantees accurate results till low temperature T/|JH| ≃ 0.01. Given the MPO representation of the thermal density matrix ρ(β) = e−βH (β ≡ 1/T), the dynamical properties at temperature T can be simulated through a successive real-time evolution. The spectral function χ′′(Q,E) ≈ − ∫tmax0 dt Im [g(Q,t)]sin(Et) · W(t/tmax) is of central interest in studying the dynamical properties at finite temperature, where W is the Hanning window function, t is the evolution time, and the (dynamical) correlation function is g(Q,t) = ⟨A Q(t)A† Q⟩ β, with A Q = √ 1/L ∑n S n+eiQn being the Fourier transformed spin operator. The time-dependent correlation function g(Q,t) can be obtained by tangent-space approach [43–45]. In the real-time evolution calculations, we consider a 16×2 lattice and evolution time up to tmax = 20/|JH|, with D = 4800 bond states retained. The damping rate γ is obtained through the DHO fitting of numerical results (see more details in Supplementary Information).

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript presents a detailed experimental investigation of the temperature evolution of magnon damping in the van der Waals metallic ferromagnet Fe3-xGeTe2. Using inelastic neutron scattering, the authors observe a nonmonotonic temperature dependence of the magnon linewidth, characterized by a minimum at an intermediate temperature (~90 K), with increased damping both at low and high temperatures. This behavior is attributed to the interplay between thermal fluctuations and Kondo screening.\nWhile the experimental data are of good quality, the results appear incremental in the context of extensive prior studies on this material. The key observation, nonmonotonic magnon damping, may be of interest, but it does not in itself constitute a conceptual advance sufficient for publication in a high-impact journal. Similar nonmonotonic behavior of the magnetic relaxation rate has been reported in heavy-fermion compounds as early as the 1980s. For example, Z. Phys. B 77, 95–104 (1989) and J. Magn. Magn. Mater. 74, 341–346 (1988) documented comparable temperature dependencies in Ce-based Kondo lattices. Theoretical interpretations of such behavior were also developed, including in J. Appl. Phys. 57, 3166–3168 (1985) and Phys. Rev. Lett. 54, 230–233 (1985). These precedents are not cited or discussed. If the intended contribution is to provide further evidence of Kondo physics in a 3d-electron system, this point has already been well established through multiple experimental techniques and in prior work by the same group. The manuscript does not make sufficiently clear what genuinely new insight is gained here.\nThe comparison with the FMKH model is, in my view, fundamentally flawed and overly simplistic. The model lacks direct relevance to the real material, which is a quasi-2D lattice system with multi-orbital degrees of freedom. The minimal 1D model presented here at best can serve as an illustration. It is unjustified to support a quantitative or qualitative comparison with experimental data. If the authors wish to pursue this model further, it would be more appropriate to present those results in a separate work, as has already been done in Ref. 56.\nThe choice of in-plane momentum transfer confuses me. Why did the authors choose an offset position like (−0.05, −0.05, 2) rather than the zone center at (0, 0, 2)? Is the magnon spectrum gapped at all temperatures?\nThe green data in panel 2c appear to correspond to the same scan shown in panel 3a. Why do the intensity not match?\nIn summary, I believe the results could be appropriate for publication in a more specialized journal focused on magnetism or neutron scattering, but do not currently meet the threshold for a high-impact journal.", "reviewer_2": "In this work, the authors explore the temperature-dependent behavior of magnon damping in ferromagnetic van der Waals (vdW) metal Fe3-xGeTe2, unveiling significant mechanisms underlying the interaction between ferromagnetism and the Kondo effect. Due to interactions between localized spins from the magnetic ions and the conduction electrons, the Kondo effect enhances electron-magnon scattering at lower temperatures with a logarithmic temperature dependence resistance, which transfers the spin angular momentum to increase the damping of the magnon. Thermal fluctuationsthermal fluctuations contribute to dephasing at higher temperatures, and the resistivity follows a power-law. The behavior of itinerant d electrons scattering collective magnon excitations of local moments discussed here has profound implications for the competition between Kondo screening and ferromagnetic exchange interactions. The intricate interplay suggests that the Kondo effect is strong enough to enhance electron-magnon scattering despite the ferromagnetic order below the Kondo temperature TK, which is crucial for developing advanced spintronics such as memory devices and magnetic applications.\nIn the study, the inelastic neutron scattering (INS) techniques were employed to investigate the low-energy spin waves in Fe3-xGeTe2. The observed phenomena indicate that magnon damping diverges at both low and high temperatures while demonstrating a distinct minimum at an intermediate temperature. Exploring the magnon dynamics in metallic van der Waals (vdW) ferromagnets Fe3-xGeTe2 allows for the meticulous examination of temperature-driven changes in spin wave behavior. Traditionally examined in f-electron systems (like those involving Ce or Yb ions), the Kondo effect is also critical in d-electron systems where localized d-orbital electrons interact with the conduction sea. The Kondo effect intricately alters the material's transport properties, such as electrical resistivity and thermopower. Upon cooling below $ T_C \\approx 160 $ K, the magnon damping rate $ Gamma$ initially decreases, reaching a minimum at approximately $ T^* \\approx 90 $ K. This behavior indicates reduced magnon-magnon and magnon-electron scattering in this temperature range. When Kondo interactions dominate, the magnetic moments are screened, leading to a non-magnetic ground state. The mechanism intertwining Kondo coupling and magnetic interactions in d-electron systems involves both direct exchange and indirect interactions mediated by conduction electrons. Tensor-Network Techniques model quantum many-body systems by efficiently capturing entanglement and dynamic interactions. Simulations reproduced the damping minimum and logarithmic temperature scaling, validating the accuracy of ferromagnetic Kondo-Heisenberg (FMKH) model in describing electron-magnon interactions and spin dynamics. This method reveals underlying phenomena of magnon damping and its distinctive temperature dependency.\nOverall, I would like to consider to recommend its publication in Nature Communications which has high standards and requirements if the following questions can be answered clearly and correctly, including the resistivity behavior, temperature-dependent magnon damping, and the implications of electron-magnon scattering linked to Kondo effect.\n1. The authors say \"These results shed light on the interplay between spin exchange and Kondo coupling in metallic 3d-electron systems with magnetic order, ... \" Although d-electrons are more delocalized compared to f-electrons, localized d-orbital electrons can still interact with conduction electrons to form localized moments strong enough for the Kondo effect. But how exactly direct exchange plays into the Kondo mechanism? Direct exchange between d-electrons and conduction electrons? Or direct exchange between localized d-moments?\n2. In the introduction it reads \"The van der Waals (vdW) metallic ferromagnet Fe3-xGeTe2 is one such example material.\" Considering the role of the van der Waals nature in layered material, in-plane conduction electrons interact more strongly with localized moments than out-of-plane. How does this anisotropy affects magnetic properties of Fe3-xGeTe2? Wether the in-plane conductivity enhance different damping along different crystallographic directions?\n3. In Fig. 1, when considering the canonical Doniach phase diagram for a ferromagnetic Kondo lattice, how to demonstrates the competition between Kondo and ordering? And how does that tie into the Kondo effect?\n4. In the Introduction the authors claim that ‘this resistivity behaviour aligns with observations in other d-electron systems exhibiting a heavy-fermion state’. Unlike 4f-electron systems, 3d electrons are more itinerant, which could affect the combined effects of Heisenberg exchange and Kondo interactions, how they compete or coexist? And Fe3-xGeTe2 has some stoichiometric variation, so the static disorder like vacancies or site substitutions becomes more impactful at lower temperatures. What about the role of disorder?\n5. The moments are more isolated in f-systems, so the interactions are purely via conduction electrons, while there are more direct exchange between neighboring d-orbitals, complicating the picture. The Kondo effect in d-electron systems involves both the Hund's coupling within the d-orbitals and the RKKY mediated by conduction electrons. Since crystal fields can split d-orbitals, how do crystal field effects influence this?\n6. As they mentioned the $gamma$ increases again below Td* due to the magnon-phonon interactions and new scattering channels. At lower temperatures, phonon populations decrease since phonons are thermal vibrations, wouldn't that reduce magnon-phonon scattering? If certain phonon modes are no longer thermally excited, the remaining phonons could have wavelengths that couple more effectively with magnons. The increase in damping below Td* might be due to the freezing out of some degrees of freedom. Whether the lattice parameters change with temperature, affecting the overlap between magnon and phonon modes? And why the optimal condition for magnon modes is around 90 K?\n7. They reproduced observed phenomena in the magnon damping minimum and logarithmic scaling within a ferromagnetic Kondo-Heisenberg (FMKH) model. This sophisticated approach leverages tensor-network techniques to accurately capture the complex interactions in quantum many-body systems. The logarithmic term is dominant at low temperatures, illustrating the enhanced Kondo screening effect, where localized magnetic moments interact strongly with conduction electrons, reducing magnetic scattering. The power-law term becomes significant at higher temperatures, reflecting the thermal excitation's impact on the resistivity.\nThe logarithmic scaling mentioned emerges from the interplay between the Kondo effect and ferromagnetism with specific temperature dependencies in the damping rate. They reproduced magnon damping minimum and a logarithmic scaling using tensor-network simulations. And how does the iron-deficiency in Fe3-xGeTe2 affect the Kondo effect and the Heisenberg interactions? What about the probably influences of the magnetic properties, like changing the number of localized spins or altering the conduction electron density?\n8.In page 7, The authors write \"This model consists of both itinerant electrons and local moments, reflecting the dual nature of d-orbital magnetism in the compound.\" The itinerant d electrons are interacting with the magnons through spin-flip processes, which is subject to renormalization effects analogous to those in the single-impurity Kondo problem. Considering the interplay between spin dynamics of itinerant electrons and collective magnetic excitations mediated by Kondo interactions, how the collective nature of magnons alters the scattering compared to single impurity cases? Does the exact mechanism of renormalization in this context involve more complex many-body effects than the traditional Kondo model?\n9. They suggest optimal conditions where thermal fluctuations are subdued, and magnon modes are relatively undisturbed by scattering processes. Around Td* $\\sim$ 90 K, the magnon damping rate is minimized. Below Td*, the damping rate rapidly increases again, likely due to enhanced magnon-phonon interactions and the onset of new scattering channels as the temperature approaches the low-temperature regime. Dominant near and above Td*, these interactions lead to damping reduction as cooperative magnetic ordering stabilizes. Below Td*, increased lattice vibrations contribute to enhanced magnon-phonon scattering, resulting in higher damping rates. Initial damping rate decrease aligns with reduced thermal disturbance and stable magnon propagation. The crossover near Td* marks a balance point between magnon scattering suppression and the potential onset of complex interactions. The minimum damping rate reflects an equilibrium state where magnon interactions are minimized. The Kondo effect itself has a characteristic temperature scale (Kondo temperature), how electron-magnon interactions fit in the heavy fermion systems? How renormalization shapes the balance between scattering mechanisms at different temperature regimes?\n10. Metallic ferromagnets have itinerant electrons, so the damping might be related to Stoner excitations, where electron-hole pairs are created, affecting the magnons. But as temperature increases towards the Curie temperature, the magnetic order starts to weaken, leading to more thermal fluctuations and thus more damping. Thus, how to address the magnon overdamping or the spectrum softening?\n11. In Eq. (2), the authors couple Kondo screening and magnetic ordering in d-electron systems involves spin-orbit coupling typically. The mechanism intertwining Kondo coupling and magnetic interactions both direct exchange and indirect interactions mediated by conduction electrons might affect the magnetic anisotropy of the localized moments. Since d-electrons are more extended, the hybridization with conduction electrons is stronger. Could this lead to different energy scales for the Kondo temperature? Why is the Kondo temperature higher in d-electron systems,, whereas it is typically very low in conventional f-electron systems?\nAdditional minor points:\n- The authors write \"By applying the renormalization of the Kondo coupling, similar to the single-impurity Kondo problem.\" How does the renormalization of spin-flip processes similar to the Kondo effect specifically adjust the interaction parameters?\n- As plotted in Fig. 4, can the spin-flip processes induced by Kondo coupling occur in materials other than Fe - deficient Fe3 - xGeTe2 single crystals?\n- The sentence in page 5 on \"...we consider Fe3−xGeTe2 demonstrates partial screening of local moments.\" How does the effective screening of magnetic moments specifically impact the electronic properties of the system?", "reviewer_3": "This manuscript detail an experimental study of spin waves in Fe deficient Fe3-xGeTe2 compound using inelastic neutron scattering. The experiments are complemented by numerical simulations. The authors claim to have found anomalous increase in the damping of the magnons at both high and low temperatures, displaying a minima at about 90 K, and relate this to the Kondo coupling in magnetically ordered systems. I do find this investigation of scientific interest, however, in my opinion, the findings in themselves are not impactful enough to warrant publication in Nature Communications. I would instead suggest to pursue publication in a more specialized journal. In addition, I have some concerns about the data analysis and interpretation, as detailed below, which give me some doubt about the robustness of the presented (key) results.\nThe inelastic neutron scattering (INS) study is conducted using a cold-neutron triple-axis instrument. For the type of detailed analysis and interpretation presented in this manuscript, I would definitely expect the authors to conduct a complete and thorough resolution analysis/deconvolution (resolution ellipsoid for a TAS instrument). This is especially important for the Q-dependent data-set AND because the c-axis of this sample is so large (i.e. small c*). It is also vital to know the “vertical acceptance/resolution” for the instrument. Here any “contamination” from the resolution ellipsoid touching the base of any nuclear Bragg peaks can easily affect the width of the magnon peak (i.e. the lifetime/damping). There is also no information about the sample mosaic, i.e. how well aligned are the collection of samples (the mosaicity of the total sample used in the INS experiments). Some scans about this should also be shown in the supplementary material in order to verify that the presented effect does not come from contamination of signal from nuclear Bragg or other magnetic intensities.\nI also have some questions regarding the analysis/results. In my understanding, the E0 parameter that is plotted e.g. in Fig. 2(e,f), is simply the peak center for the magnon. In this case, looking e.g. at the actual data of Fig. 2(c) and comparing it to results in Fig. 2(e), it does not seem to match. For instance, the blue 4 K data shows a peak center around 0.3 meV in Fig. 2(c), however, in Fig. 2(e) the blue point at 4 K is around 1 meV. Further, in Fig. 2(e) there is a blue point at Tc however, no red point is shown, but the red line indicate that there might be a point very high up (outside the graph). In fact, the main point indicating that there is a strong increase in the damping for higher temperatures is the 140 K point in Fig. 2(e). However, looking at the actual fits in Fig. 2(c), for me it is not evident that this peak is much broader than the green 90 K data. Finally, the 4 K peak in Fig. 2(c) should according to plot in Fig. 2(e) be broader than all other peaks... it definitely does not look like this is the case (which also matches the fact that center E0 seems to be wrong). All these points makes me partly doubt the final presented (key) results about divergence of magnon damping at both high and low temperature.\nFinally, I would have liked to see some information about the sample quality also in this manuscript (and/or supplementary material). It would be of vital importance to show that this Fe deficient sample is not phase separated. I also did not find any information about this in Reference 31 (which present data from the same sample).\nIn summary, I cannot recommend publication of the current manuscript in Nature Communications."}, "author_response": "Response to the Reviewers\nWe sincerely thank all three reviewers for their time, effort, and constructive\nevaluations of our manuscript. We are grateful that they all found the topic of our work\nto be of interest and acknowledged the quality of our experimental data. Reviewer 2\nprovided a particularly positive and detailed assessment, emphasizing the significance\nof our findings for understanding the interplay between Kondo screening and\nferromagnetic order in d-electron systems, as well as their potential implications for\nspintronics applications. Reviewer 2 explicitly indicated a willingness to recommend\npublication in Nature Communications once several technical points are clarified,\nwhich we will fully address below. Reviewer 3 acknowledged the interest of our work\nand posed questions regarding data analysis and interpretation. While this reviewer\nexpressed a personal view that the impact might be better suited to a specialized journal,\nthe concerns raised are primarily technical in nature, and we believe they can be\nresolved through clarification and additional discussion in the revised manuscript.\nThe main concern regarding novelty arises from Reviewer 1, who cited earlier heavy-\nfermion studies from the 1980s reporting nonmonotonic temperature dependence of\nmagnetic relaxation rates. We appreciate for pointing out these interesting references;\nhowever, we respectfully note that the works cited by Reviewer 1 investigated\nmomentum-independent local 4f-moment fluctuations in paramagnetic or weakly\nordered systems, typically measured by NMR or quasielastic neutron scattering. In\ncontrast, our study presents measurement of momentum-resolved damping of collective\nexcitations (magnons) in a 3d-electron Kondo lattice with long-range magnetic order,\nusing inelastic neutron scattering. This finding, observed for the first time, is ascribed\nto a qualitatively different physical mechanism, supported by quantitative scaling\nbehavior, clear momentum dependence, and tensor-network simulations of the Kondo-\nHeisenberg model. These distinctions are elaborated in detail in our point-by-point\nresponse.\nWe value the constructive feedback from all reviewers, and in the revised manuscript\nwe address all technical and interpretational questions in full. The changes are marked\nin red, and a summary of all modifications is appended after the response to the\nreviewers. We believe these clarifications resolve the concerns the reviewers had and\ndemonstrate that our work meets the high standards of Nature Communications in both\nconceptual significance and broad interest.\n1 / 22\n-------------------------------------------------------------------------------------------------------\nReviewer #1\n-------------------------------------------------------------------------------------------------------\nComment:\nThis manuscript presents a detailed experimental investigation of the temperature\nevolution of magnon damping in the van der Waals metallic ferromagnet Fe3-xGeTe2.\nUsing inelastic neutron scattering, the authors observe a nonmonotonic temperature\ndependence of the magnon linewidth, characterized by a minimum at an intermediate\ntemperature (~90 K), with increased damping both at low and high temperatures. This\nbehavior is attributed to the interplay between thermal fluctuations and Kondo\nscreening.\nWhile the experimental data are of good quality, the results appear incremental in the\ncontext of extensive prior studies on this material. The key observation, nonmonotonic\nmagnon damping, may be of interest, but it does not in itself constitute a conceptual\nadvance sufficient for publication in a high-impact journal. Similar nonmonotonic\nbehavior of the magnetic relaxation rate has been reported in heavy-fermion compounds\nas early as the 1980s. For example, Z. Phys. B 77, 95–104 (1989) and J. Magn. Magn.\nMater. 74, 341–346 (1988) documented comparable temperature dependencies in Ce-\nbased Kondo lattices. Theoretical interpretations of such behavior were also developed,\nincluding in J. Appl. Phys. 57, 3166–3168 (1985) and Phys. Rev. Lett. 54, 230–233\n(1985). These precedents are not cited or discussed. If the intended contribution is to\nprovide further evidence of Kondo physics in a 3d-electron system, this point has\nalready been well established through multiple experimental techniques and in prior\nwork by the same group. The manuscript does not make sufficiently clear what\ngenuinely new insight is gained here.\nThe comparison with the FMKH model is, in my view, fundamentally flawed and\noverly simplistic. The model lacks direct relevance to the real material, which is a quasi-\n2D lattice system with multi-orbital degrees of freedom. The minimal 1D model\npresented here at best can serve as an illustration. It is unjustified to support a\nquantitative or qualitative comparison with experimental data. If the authors wish to\npursue this model further, it would be more appropriate to present those results in a\nseparate work, as has already been done in Ref. 56.\nResponse:\nWe thank the reviewer for the careful reading of our manuscript and for recognizing the\nquality of our experimental data. We also appreciate the opportunity to clarify both the\nnovelty of our work and the role of the theoretical modeling.\nThe main concern regarding novelty arises from the comparison to a set of heavy-\nfermion studies from the 1980s (e.g., Ce-based compounds) cited by the reviewer,\nwhich reported a nonmonotonic temperature dependence of the magnetic relaxation rate\nγ(T). While being grateful to know these references, we emphasize that, although the\n2 / 22\nform of γ(T) appears similar at first glance, the physical systems, spin dynamics,\nunderlying mechanism, and experimental observable are all distinct.\nThe earlier heavy-fermion studies examined momentum-independent single 4f-moment\nfluctuations in paramagnetic or weakly ordered systems, probed by NMR or\nquasielastic neutron scattering. The extracted relaxation rates reflected the lifetime of\nsingle spin fluctuations, typically interpreted within conventional single-impurity\nKondo models.\nIn sharp contrast, our work presents the first momentum-resolved measurement of a\nnovel damping phenomenon for the collective excitations of ordered local moments\n(magnons, as opposed to single spin), probed by inelastic neutron scattering. The\nextracted damping rates reflect the lifetime of magnons, governed by a mechanism\ninvolving Kondo-coupling induced magnon-electron scattering, distinct from prior\nobservations in 4f-based systems. Our work thus opens a new direction for probing\nKondo coupling through collective spin excitations.\nFor clarity, the essential distinctions are summarized below in the table:\nAspect Prior Studies This Work\nMagnetic state Paramagnetic or weakly ordered Long-range ordered\nSpin dynamics Single spin fluctuations Collective excitations (magnons)\nProbing\nQuasielastic neutron scattering / NMR Inelastic neutron scattering\ntechnique\nPhysical\nRelaxation of a single spin Damping of magnons\nobservable\nQ dependence Absent Present\nTheoretical\nSingle-impurity Kondo model Kondo-Heisenberg lattice model\nmodel\nPhysical Conventional single-impurity Kondo Kondo-coupling induced\nmechanism scattering magnon-electron scattering\nWe further substantiate our conclusions by:\n(i) establishing a quantitative scaling law γ(T) ∼ ln(T*/T) + power law;\n(ii) demonstrating momentum dependence along both in-plane and out-of-plane\ndirections; and\n(iii) validating the behavior via tensor-network simulations of the ferromagnetic\nKondo-Heisenberg model---applied here for the first time in this context.\nWe thank the reviewer for pointing out the relevant theoretical studies based on\nAnderson-type models [e.g., Phys. Rev. Lett. 54, 230 (1985)]. Regarding theoretical\nmodeling, our work represents a distinct and essential step forward for understanding\nthe metallic ferromagnet Fe GeTe . Unlike previous approaches that focus on electron\n3-x 2\nscattering from a single Ce impurity, our model successfully captures the interplay\nbetween itinerant electrons and collective spin excitations---specifically, ferromagnetic\n3 / 22\nspin waves.\nThis framework is crucial for understanding magnon damping in Fe GeTe . The 1D\n3-x 2\nFMKH model we employed is indeed a minimal model designed to capture the essential\nphysics of an FM Kondo lattice, namely, the competition between ferromagnetic\nexchange and Kondo coupling, which qualitatively reproduces the damping minimum\nobserved experimentally. Our intention was not to conduct a quantitative comparison\nbetween the minimal model and the material, but rather to demonstrate that the\nnonmonotonic γ(T) emerges within a controlled many-body framework that\nincorporates both collective spin excitations and Kondo coupling.\nNevertheless, we fully agree with the reviewer that a more quantitative comparison\nnecessitates a 2D, multi-orbital model to accurately capture the quasi-2D nature of Fe\n3-\nGeTe . While such an extension is beyond the scope of the present manuscript due to\nx 2\nthe significant challenge of converging real-time dynamics in 2D, we view this as a\nstrong incentive for the further development of tensor-network techniques, representing\nan important direction for future work.\nIn response to the reviewer’s comments, we have cited relevant literature and expanded\nthe discussion in the revised manuscript to clarify these distinctions. We have also\nprovided additional explanation regarding the choice of the 1D model.\nComment:\nThe choice of in-plane momentum transfer confuses me. Why did the authors choose\nan offset position like (−0.05, −0.05, 2) rather than the zone center at (0, 0, 2)? Is the\nmagnon spectrum gapped at all temperatures?\nResponse:\nWe thank the reviewer for this question and apologize for the confusion. The offset\nposition was chosen in order to track the temperature evolution of magnon excitations,\nwhere a well-defined inelastic magnon peak can be clearly resolved (see Figs. 2a and\n2b in the main text). By contrast, at the zone center (0, 0, 2), no sizable spin gap is\nobserved; instead, the spectra are dominated by quasielastic scattering. This indicates\nthat the single-ion anisotropy in the bulk is relatively weak compared with the\nmonolayer limit. In addition, diffusive fluctuations induced by itinerant electrons may\npartially fill the tiny spin gap (see Supplementary Fig. 8 and the corresponding\ndiscussion in Ref. 31).\nThe chosen offset positions in-plane (−0.05, −0.05, 2) and out-of-plane (0, 0, 1.7), also\nallow us to probe the anisotropic damping behavior along these two directions [Figs.\n2e and 2f]. To ensure that our conclusions are not dependent on a specific momentum\nchoice, we further examined the momentum dependence of magnon damping (Fig. 3).\nThese results confirm that the nonmonotonic temperature dependence of magnon\n4 / 22\ndamping is a robust and universal feature, independent of momentum.\nIn response to the reviewer’s suggestion, we have added notes in the revised manuscript\nto clarify why offset positions were used instead of the exact zone center.\nComment:\nThe green data in panel 2c appear to correspond to the same scan shown in panel 3a.\nWhy do the intensity not match?\nResponse:\nWe thank the reviewer for the careful examination of our work. The data in Figs. 2 and\n3 were collected in different experimental runs with slightly different instrument\nsettings. In the first batch (Fig. 2), a cooled Be filter was placed after the sample, while\nin the second batch (Fig. 3) the filter was removed resulting in higher detected neutron\nintensity. As a result, the overall intensity in Fig. 3 appears stronger. We have added a\nnote in the revised manuscript to clarify this point.\nComment:\nIn summary, I believe the results could be appropriate for publication in a more\nspecialized journal focused on magnetism or neutron scattering, but do not currently\nmeet the threshold for a high-impact journal.\nResponse:\nWe sincerely thank the reviewer for the careful evaluation of our work. We respectfully\ndisagree with the assessment that the results do not meet the threshold for a high-impact\njournal. In the revised manuscript, we have clarified the key distinctions from prior\nheavy-fermion studies and added relevant citations and discussion. Here, we unveil a\nnovel damping phenomenon for the collective excitations in a metallic ferromagnet\n(as opposed to single spin fluctuations), which is attributed to a new mechanism\ninvolving Kondo-coupling induced magnon-electron scattering, distinct from prior\nobservations in 4f-based systems described by the Anderson-type impurity model. We\nbelieve that magnon damping is a ubiquitous phenomenon in metallic ferromagnets and\nother magnetically ordered metals, and therefore serves as a sensitive probe of Kondo\ncouplings in a broad range of systems. With these revisions, we have further clarified\nthe novelty and significance of our work, and respectfully request that the reviewer re-\nevaluate the manuscript for publication in Nature Communications.\n5 / 22\n-------------------------------------------------------------------------------------------------------\nReviewer #2\n-------------------------------------------------------------------------------------------------------\nComment:\nIn this work, the authors explore the temperature-dependent behavior of magnon\ndamping in ferromagnetic van der Waals (vdW) metal Fe3-xGeTe2, unveiling\nsignificant mechanisms underlying the interaction between ferromagnetism and the\nKondo effect. Due to interactions between localized spins from the magnetic ions and\nthe conduction electrons, the Kondo effect enhances electron-magnon scattering at\nlower temperatures with a logarithmic temperature dependence resistance, which\ntransfers the spin angular momentum to increase the damping of the magnon. Thermal\nfluctuations contribute to dephasing at higher temperatures, and the resistivity follows\na power-law. The behavior of itinerant d electrons scattering collective magnon\nexcitations of local moments discussed here has profound implications for the\ncompetition between Kondo screening and ferromagnetic exchange interactions. The\nintricate interplay suggests that the Kondo effect is strong enough to enhance electron-\nmagnon scattering despite the ferromagnetic order below the Kondo temperature TK,\nwhich is crucial for developing advanced spintronics such as memory devices and\nmagnetic applications.\nIn the study, the inelastic neutron scattering (INS) techniques were employed to\ninvestigate the low-energy spin waves in Fe3-xGeTe2. The observed phenomena\nindicate that magnon damping diverges at both low and high temperatures while\ndemonstrating a distinct minimum at an intermediate temperature. Exploring the\nmagnon dynamics in metallic van der Waals (vdW) ferromagnets Fe3-xGeTe2 allows\nfor the meticulous examination of temperature-driven changes in spin wave behavior.\nTraditionally examined in f-electron systems (like those involving Ce or Yb ions), the\nKondo effect is also critical in d-electron systems where localized d-orbital electrons\ninteract with the conduction sea. The Kondo effect intricately alters the material's\ntransport properties, such as electrical resistivity and thermopower. Upon cooling below\n$ T_C \\approx 160 $ K, the magnon damping rate $ Gamma$ initially decreases,\nreaching a minimum at approximately $ T^* \\approx 90 $ K. This behavior indicates\nreduced magnon-magnon and magnon-electron scattering in this temperature range.\nWhen Kondo interactions dominate, the magnetic moments are screened, leading to a\nnon-magnetic ground state. The mechanism intertwining Kondo coupling and magnetic\ninteractions in d-electron systems involves both direct exchange and indirect\ninteractions mediated by conduction electrons. Tensor-Network Techniques model\nquantum many-body systems by efficiently capturing entanglement and dynamic\ninteractions. Simulations reproduced the damping minimum and logarithmic\ntemperature scaling, validating the accuracy of ferromagnetic Kondo-Heisenberg\n(FMKH) model in describing electron-magnon interactions and spin dynamics. This\nmethod reveals underlying phenomena of magnon damping and its distinctive\ntemperature dependency.\n6 / 22\nOverall, I would like to consider to recommend its publication in Nature\nCommunications which has high standards and requirements if the following questions\ncan be answered clearly and correctly, including the resistivity behavior, temperature-\ndependent magnon damping, and the implications of electron-magnon scattering linked\nto Kondo effect.\nResponse:\nWe sincerely thank the reviewer for the thorough and insightful evaluation of our work.\nWe greatly appreciate the recognition of the novelty and importance of our study---the\nobservation of a temperature-dependent magnon damping minimum in the metallic van\nder Waals ferromagnet Fe GeTe and its connection to the Kondo effect in a d-electron\n3-x 2\nsystem. We are encouraged by the reviewer’s positive assessment that our work could\nmeet the high standards of Nature Communications provided that the key questions are\naddressed.\nIn the revised manuscript, we have carefully clarified the three major issues highlighted\nby the reviewer: (i) the resistivity behavior with incoherent-coherent crossover, (ii) the\ntemperature-dependent magnon damping with its characteristic scaling and minimum,\nand (iii) the implications of electron-magnon scattering in the presence of the Kondo\neffect. We show that our experimental results are consistent with transport\nmeasurements, theoretical modeling, and the established understanding of Kondo\nphysics, thereby strengthening the central conclusions of our study.\nComment:\n1. The authors say \"These results shed light on the interplay between spin exchange and\nKondo coupling in metallic 3d-electron systems with magnetic order, ... \" Although d-\nelectrons are more delocalized compared to f-electrons, localized d-orbital electrons\ncan still interact with conduction electrons to form localized moments strong enough\nfor the Kondo effect. But how exactly direct exchange plays into the Kondo mechanism?\nDirect exchange between d-electrons and conduction electrons? Or direct exchange\nbetween localized d-moments?\nResponse:\nWe thank the reviewer for this insightful question. We think there is strong direct\nferromagnetic exchange coupling between localized d-moments in Fe GeTe , which\n3-x 2\ngives rise to long-range ferromagnetic order. The direct exchange interaction comes\nfrom the overlap of the outermost 3d orbitals of Fe ions, which is strong and leads to a\nrelatively high magnetic transition temperature. In contrast, in conventional f-electron\nKondo-lattice systems, the inner f electrons are effectively screened by conduction\nelectrons in outer orbitals, so the localized f-moments interact mainly via the\n(Ruderman-Kittel-Kasuya-Yosida) RKKY interaction, which is weaker and usually\nresults in a lower transition temperature and antiferromagnetic order.\n7 / 22\nIn our system, the observed behavior reflects the competition between strong direct\nferromagnetic exchange among localized d moments and antiferromagnetic Kondo\ncoupling to itinerant electrons. To capture this situation, we developed an effective\nferromagnetic Kondo-Heisenberg lattice model that incorporates both interactions and\nexplains the observed magnon damping.\nIn the revised manuscript, we have emphasized the dominant role of direct\nferromagnetic exchange in stabilizing the magnetic order, and explicitly compared this\nwith the RKKY-mediated case in f-electron systems.\nComment:\n2. In the introduction it reads \"The van der Waals (vdW) metallic ferromagnet Fe3-\nxGeTe2 is one such example material.\" Considering the role of the van der Waals nature\nin layered material, in-plane conduction electrons interact more strongly with localized\nmoments than out-of-plane. How does this anisotropy affects magnetic properties of\nFe3-xGeTe2? Wether the in-plane conductivity enhance different damping along\ndifferent crystallographic directions?\nResponse:\nWe thank the reviewer for this insightful question. Indeed, the van der Waals nature of\nFe GeTe leads to an anisotropic electronic structure: the in-plane conduction\n3-x 2\nelectrons have higher mobility and stronger overlap with the localized d moments\ncompared to the out-of-plane direction. Consequently, the Kondo coupling is stronger\nin-plane, naturally leading to larger magnon damping along the in-plane direction than\nalong the out-of-plane direction. This anisotropic damping behavior is consistent with\nour experimental observations in Figs. 2e and 2f, where the in-plane γ is systematically\nhigher. Moreover, the stronger in-plane Kondo screening reduces the effective\nintralayer exchange between local moments, contributing to the observed magnon\nsoftening at low temperatures, as shown in Figs. 2e and 3g. We have highlighted this\ndiscussion in the last paragraph of the section “Damping rate minimum and logarithmic\nscaling” of revised manuscript.\nComment:\n3. In Fig. 1, when considering the canonical Doniach phase diagram for a ferromagnetic\nKondo lattice, how to demonstrates the competition between Kondo and ordering? And\nhow does that tie into the Kondo effect?\nResponse:\nWe thank the reviewer for this insightful question. In contrast to the original Doniach\nphase diagram, where the long-range magnetic order is due to indirect RKKY\ninteraction, here we introduce an FM Heisenberg exchange between the local d-electron\nmoments. The resulting phase diagram is Doniach-like, as it features a competition\n8 / 22\nbetween this FM order and the Kondo singlet formation, but it is not identical due to\nthe presence of this additional direct exchange. When the FM exchange dominates, the\nsystem exhibits long-range FM order; when the Kondo coupling dominates, the local\nmoments are screened, leading to a heavy Fermi liquid. These two regimes are\nseparated by a quantum critical point, as illustrated schematically in Fig. 1e. More\ndetailed numerical results and analysis of this phase diagram are presented in our\nseparate theoretical work (now Ref. 60).\nComment:\n4. In the Introduction the authors claim that ‘this resistivity behaviour aligns with\nobservations in other d-electron systems exhibiting a heavy-fermion state’. Unlike 4f-\nelectron systems, 3d electrons are more itinerant, which could affect the combined\neffects of Heisenberg exchange and Kondo interactions, how they compete or coexist?\nAnd Fe3-xGeTe2 has some stoichiometric variation, so the static disorder like\nvacancies or site substitutions becomes more impactful at lower temperatures. What\nabout the role of disorder?\nResponse:\nWe thank the reviewer for this insightful question. Unlike f-electron systems, the 3d\nelectrons in Fe GeTe are more itinerant. From the resistivity perspective, this\n3-x 2\nmanifests as a slope change around the incoherent-to-coherent crossover, rather than a\npronounced hump typical of f-electron heavy-fermion systems, as illustrated in Fig. 1d.\nRegarding the competition of interactions, the itinerancy enhances conduction-\nelectron-mediated Kondo coupling while still allowing strong direct ferromagnetic\nexchange between localized d moments. Consequently, the system exhibits a\ncompetition between long-range ferromagnetic order and Kondo screening, which is\ncaptured by our ferromagnetic Kondo-Heisenberg lattice model. The stronger\nitinerancy shifts the relevant energy scales and temperature range over which Kondo\nand exchange interactions influence magnon damping, leading to the observed\nlogarithmic and power-law temperature dependence.\nRegarding disorder, Fe deficiency and site substitutions can locally modify Kondo\nhybridization and exchange interactions. For example, Ref. 30 reports anomalous\nenhancement of Kondo screening at Kondo-hole sites associated with local Fe\nvacancies in Fe GeTe . However, our bulk measurements, including inelastic neutron\n3 2\nscattering, probe volume-averaged properties that are insensitive to such local\nimperfections. Overall, Fe deficiencies reduce both ferromagnetic and Kondo couplings,\nlowering T and T* compared to Fe-rich compounds (e.g., T ≈ 160 K vs. 230 K, T* ≈\nC C\nKondo couplings. We emphasize that our theoretical model does not explicitly\nincorporate iron vacancies; rather, the iron deficiency should be viewed as an\nexperimental tuning knob that modifies the relative strengths of the two competing\ninteractions. In the present work we have focused on a single composition, but a\nsystematic investigation of how Fe vacancies influence both the Kondo effect and\nHeisenberg interactions would indeed be a very interesting direction for future studies.\n9 / 22\nIn the revised manuscript, we have added discussions of the competition between\nHeisenberg exchange and Kondo interactions in 3d-electron systems in the section\n“FMKH model and spin-flip scattering mechanism”. We have also added some\ndiscussions on the effects of Fe vacancy.\nComment:\n5. The moments are more isolated in f-systems, so the interactions are purely via\nconduction electrons, while there are more direct exchange between neighboring d-\norbitals, complicating the picture. The Kondo effect in d-electron systems involves both\nthe Hund's coupling within the d-orbitals and the RKKY mediated by conduction\nelectrons. Since crystal fields can split d-orbitals, how do crystal field effects influence\nthis?\nResponse:\nWe thank the reviewer for raising this interesting point. In 3d-electron systems, crystal\nfield effects can indeed split the d-orbitals and modify the local spin configuration,\nwhich in turn may influence the effective Kondo coupling and the spectrum of low-\nlying spin excitations. In Fe GeTe , however, the dominant direct ferromagnetic\n3-x 2\nexchange between localized d moments appears to govern the overall magnetic order,\nand our observed temperature-dependent magnon damping is primarily determined by\nthe competition between Kondo coupling and exchange interactions. Consistently, in\nthe measured magnetic excitation spectra (up to 100 meV, in Ref. 31), we do not observe\nany discernible crystal field excitations, suggesting that crystal field effects are\nrelatively weak in this system. Therefore, while crystal field effect could introduce\nadditional refinements, it is not expected to qualitatively alter the nonmonotonic\ndamping behavior reported here.\nA more systematic investigation of crystal field effects would certainly be valuable, but\nit lies beyond the scope of the present work. We expect that future studies could shed\nfurther light on this intriguing aspect.\nComment:\n6. As they mentioned the $gamma$ increases again below Td* due to the magnon-\nphonon interactions and new scattering channels. At lower temperatures, phonon\npopulations decrease since phonons are thermal vibrations, wouldn't that reduce\nmagnon-phonon scattering? If certain phonon modes are no longer thermally excited,\nthe remaining phonons could have wavelengths that couple more effectively with\nmagnons. The increase in damping below Td* might be due to the freezing out of some\ndegrees of freedom. Whether the lattice parameters change with temperature, affecting\nthe overlap between magnon and phonon modes? And why the optimal condition for\n10 / 22\nmagnon modes is around 90 K?\nResponse:\nWe thank the reviewer for this insightful question. First, we would like to clarify that\nour work attributes the additional damping channel to Kondo-driven magnon-electron\nscattering, rather than magnon-phonon coupling. In itinerant magnetic systems, it is\nmore natural to consider scattering between spin excitations and itinerant electrons,\nand our ferromagnetic Kondo-Heisenberg lattice model reproduces both the scaling\nbehavior and the nonmonotonic damping minimum. By contrast, spin-lattice coupling,\nwhich can broaden spin waves via fluctuations of exchange interactions caused by\nlattice vibrations [see Nat. Commun. 13, 4037 (2022)], should be reduced at low\ntemperatures as lattice displacements decrease. This behavior is distinct from the\nKondo-driven mechanism observed in Fe GeTe .\n3-x 2\nBoth thermal fluctuations and Kondo scattering can damp magnons. Thermal effects\ndominate at high temperatures, while Kondo scattering becomes increasingly relevant\nat low temperatures. Their competition naturally leads to an intermediate temperature\nwhere the damping is minimized. Intriguingly, this occurs near ~90 K, which coincides\nwith the incoherent-coherent crossover observed in the resistivity (Fig. 1d). This\ntemperature marks the onset of hybridization between localized and itinerant electronic\nstates, accompanied by an enlargement of the Fermi surface volume (Ref. 29). From\nthe perspective of magnons, this crossover provides additional decay channels,\nenhancing magnon damping. This explains why the optimal condition for long-lived\nmagnon modes occurs around 90 K. We have discussed this point in the beginning of\nthe section “FMKH model and spin-flip scattering mechanism”.\nComment:\n7. They reproduced observed phenomena in the magnon damping minimum and\nlogarithmic scaling within a ferromagnetic Kondo-Heisenberg (FMKH) model. This\nsophisticated approach leverages tensor-network techniques to accurately capture the\ncomplex interactions in quantum many-body systems. The logarithmic term is\ndominant at low temperatures, illustrating the enhanced Kondo screening effect, where\nlocalized magnetic moments interact strongly with conduction electrons, reducing\nmagnetic scattering. The power-law term becomes significant at higher temperatures,\nreflecting the thermal excitation's impact on the resistivity.\nThe logarithmic scaling mentioned emerges from the interplay between the Kondo\neffect and ferromagnetism with specific temperature dependencies in the damping rate.\nThey reproduced magnon damping minimum and a logarithmic scaling using tensor-\nnetwork simulations. And how does the iron-deficiency in Fe3-xGeTe2 affect the\nKondo effect and the Heisenberg interactions? What about the probably influences of\nthe magnetic properties, like changing the number of localized spins or altering the\nconduction electron density?\n11 / 22\nResponse:\nWe thank the reviewer for the summary and praise of our theoretical modelling, and for\nthe important question on iron deficiencies. As pointed out, iron deficiency in Fe\n3-\nGeTe simultaneously tunes the density of localized spins and the itinerant electron\nx 2\nconcentration. Missing Fe sites reduce the net magnetic moment and weaken the\nferromagnetic exchange, while the accompanying change in band filling alters the\ncarrier density that controls the Kondo hybridization strength. Experimentally, this dual\neffect is reflected in the reduction of both T and the crossover temperature T* in Fe\nC 3-\nGeTe compared to the stoichiometric compound (T ≈ 160 K vs. 230 K, T* ≈\nx 2 C\nKondo couplings. We emphasize that our theoretical model does not explicitly\nincorporate iron vacancies; rather, the iron deficiency should be viewed as an\nexperimental tuning knob that modifies the relative strengths of the two competing\ninteractions. In the present work we have focused on a single composition, but a\nsystematic investigation of how Fe vacancies influence both the Kondo effect and\nHeisenberg interactions would indeed be a very interesting direction for future studies.\nComment:\n8.In page 7, The authors write \"This model consists of both itinerant electrons and local\nmoments, reflecting the dual nature of d-orbital magnetism in the compound.\" The\nitinerant d electrons are interacting with the magnons through spin-flip processes,\nwhich is subject to renormalization effects analogous to those in the single-impurity\nKondo problem. Considering the interplay between spin dynamics of itinerant electrons\nand collective magnetic excitations mediated by Kondo interactions, how the collective\nnature of magnons alters the scattering compared to single impurity cases? Does the\nexact mechanism of renormalization in this context involve more complex many-body\neffects than the traditional Kondo model?\nResponse:\nWe thank the reviewer for this insightful question. The key distinction from the single-\nimpurity Kondo problem lies in the collective nature of magnons. In the traditional case,\nKondo renormalization reflects the repeated spin-flip scattering between itinerant\nelectrons and a single localized spin, leading to screening at low temperatures. In\ncontrast, in Fe GeTe the localized spins are ferromagnetically ordered, and the\n3-x 2\nrelevant processes involve scattering between itinerant electrons and the collective\nexcitations (magnons) of this ordered state, which is naturally treated within the\nferromagnetic Kondo-Heisenberg lattice framework.\nThis collective character alters the renormalization in two important ways. First, the\nitinerant electrons couple coherently to magnon modes, giving rise to momentum-\ndependent renormalization of the magnon spectrum rather than purely local screening.\nSecond, these processes feed back into the magnon self-energy, enabling anomalous\n12 / 22\nmagnon decay. By incorporating the renormalization of the Kondo coupling (Fig. 4b)-\n--formally similar to the impurity case---we reproduce the observed damping minimum\nand logarithmic scaling behavior.\nThus, while the analogy with the impurity Kondo effect provides useful intuition, the\nrenormalization mechanism here involves more complex many-body physics,\ncombining itinerant electron-magnon scattering with lattice-type Kondo hybridization.\nOur tensor-network calculations are specifically designed to capture both (i) the Kondo-\ntype renormalization of the coupling between itinerant electrons and localized spins,\nand, (ii) the resulting self-energy corrections to magnons, highlighting the essential\ncompetition between ferromagnetic exchange and Kondo coupling. To make this\ndistinction clearer, we have added explicit discussion in the revised manuscript to\nemphasize how our model goes beyond the single-impurity scenario.\nComment:\n9. They suggest optimal conditions where thermal fluctuations are subdued, and\nmagnon modes are relatively undisturbed by scattering processes. Around Td*\n$\\sim$ 90 K, the magnon damping rate is minimized. Below Td*, the damping rate\nrapidly increases again, likely due to enhanced magnon-phonon interactions and the\nonset of new scattering channels as the temperature approaches the low-temperature\nregime. Dominant near and above Td*, these interactions lead to damping reduction as\ncooperative magnetic ordering stabilizes. Below Td*, increased lattice vibrations\ncontribute to enhanced magnon-phonon scattering, resulting in higher damping rates.\nInitial damping rate decrease aligns with reduced thermal disturbance and stable\nmagnon propagation. The crossover near Td* marks a balance point between magnon\nscattering suppression and the potential onset of complex interactions. The minimum\ndamping rate reflects an equilibrium state where magnon interactions are minimized.\nThe Kondo effect itself has a characteristic temperature scale (Kondo temperature),\nhow electron-magnon interactions fit in the heavy fermion systems? How\nrenormalization shapes the balance between scattering mechanisms at different\ntemperature regimes?\nResponse:\nWe thank the reviewer for the detailed comments. We would like to clarify that the\nincrease of magnon damping below T * ~90 K is not primarily due to magnon-phonon\nd\ninteractions, but instead originates from enhanced Kondo-driven electron-magnon\nscattering, as we explained in response to Comment #6. As the temperature decreases,\nthe hybridization between localized 𝑑-moments and conduction electrons becomes\nstronger, which reshapes the Fermi surface and simultaneously opens an additional\ndecay channel for magnons.\nThe resulting competition between two scattering mechanisms---Kondo coupling that\ndominates at low temperatures, and thermal fluctuations that dominate at high\n13 / 22\ntemperatures---naturally produces the observed damping minimum around T * ~90 K.\nd\nThis crossover coincides with the incoherent-to-coherent transition in the resistivity,\nfurther supporting the Kondo interpretation. Within this framework, the low-\ntemperature logarithmic scaling of magnon damping directly reflects Kondo screening,\nwhereas the high-temperature power-law behavior is governed by conventional thermal\nscattering.\nThus, unlike the Kondo effect in heavy-fermion systems which primarily involves with\nitinerant electrons and single-ion magnetic moments, here the Kondo effect manifests\nthrough electron-magnon interactions, and defines a characteristic temperature scale\nwhere the renormalization of the Kondo coupling shapes the balance between different\nscattering channels. We have expanded the discussion in the revised manuscript to make\nthis mechanism and its temperature dependence clearer.\nComment:\n10. Metallic ferromagnets have itinerant electrons, so the damping might be related to\nStoner excitations, where electron-hole pairs are created, affecting the magnons. But as\ntemperature increases towards the Curie temperature, the magnetic order starts to\nweaken, leading to more thermal fluctuations and thus more damping. Thus, how to\naddress the magnon overdamping or the spectrum softening?\nResponse:\nWe thank the reviewer for this question. In metallic ferromagnets, magnon damping can\nhave contributions from Stoner excitations (electron-hole pair creation), which are\nmore relevant at higher energies. This has been extensively discussed in our previous\nwork (Ref. 31). However, in Fe GeTe , the observed low-energy magnon\n3-x 2\noverdamping at both low and high temperatures, and the nonmonotonic temperature\ndependence of the damping constant γ, cannot be explained by Stoner processes alone.\nAt high temperatures approaching T , thermal fluctuations increase, contributing to\nC\nmagnon broadening and the observed power-law behavior of γ. At low temperatures,\nKondo-driven electron-magnon scattering dominates, leading to additional magnon\ndecay channels and the characteristic logarithmic-law scaling of γ.\nRegarding magnon softening, for the out-of-plane direction where Kondo coupling is\nweaker, softening near T , is primarily due to thermal fluctuations (Figs. 2f and 3g).\nC\nFor the in-plane direction, stronger Kondo coupling reduces the effective intralayer\nexchange between local moments at low temperatures, leading to pronounced softening\nthat exceeds the effect of thermal fluctuations, as shown in Figs. 2e and 3g. This\nanisotropic behavior highlights the essential role of Kondo interactions in shaping both\nthe damping and dispersion of magnons in Fe GeTe , as we have discussed in the\n3-x 2\nrevised manuscript.\n14 / 22\nComment:\n11. In Eq. (2), the authors couple Kondo screening and magnetic ordering in d-electron\nsystems involves spin-orbit coupling typically. The mechanism intertwining Kondo\ncoupling and magnetic interactions both direct exchange and indirect interactions\nmediated by conduction electrons might affect the magnetic anisotropy of the localized\nmoments. Since d-electrons are more extended, the hybridization with conduction\nelectrons is stronger. Could this lead to different energy scales for the Kondo\ntemperature? Why is the Kondo temperature higher in d-electron systems,, whereas it\nis typically very low in conventional f-electron systems?\nResponse:\nWe thank the reviewer for this question. As noted, the key reason why the Kondo\ntemperature is higher in d-electron systems such as Fe GeTe compared to\n3-x 2\nconventional f-electron systems is the more extended nature of the 3d orbitals. This\nleads to stronger hybridization with conduction electrons, enhancing the Kondo\ncoupling and thus increasing Kondo temperature. In contrast, localized f electrons\nhybridize weakly with conduction electrons, resulting in much lower Kondo\ntemperatures.\nWhile spin-orbit coupling can influence magnetic anisotropy, our present ferromagnetic\nKondo-Heisenberg lattice model focuses on the competition between direct\nferromagnetic exchange and Kondo coupling, and does not explicitly include spin-\norbit-mediated effects. The essential mechanism of our model determining Kondo\ntemperature is therefore the strength of the itinerant-localized electron hybridization.\nWe appreciate the reviewer’s insightful questions regarding the broader implications\nfor magnetic anisotropy and Kondo energy scales in d-electron systems. While some of\nour discussion suggests plausible scenarios, a complete understanding of these aspects\nrequires further dedicated studies beyond the scope of the present work. We look\nforward to addressing these issues in future investigations.\nComment:\nAdditional minor points:\n- The authors write \"By applying the renormalization of the Kondo coupling, similar to\nthe single-impurity Kondo problem.\" How does the renormalization of spin-flip\nprocesses similar to the Kondo effect specifically adjust the interaction parameters?\nResponse:\nWe thank the reviewer for this insightful question. In our modeling, the Kondo coupling\nis renormalized as J →J ren=J [1+J g(ϵ )log(Λ/T)] to capture the observed damping\nK K K K F\nminimum and logarithmic temperature scaling (see details in Supplementary Materials).\nWhile this renormalization form is formally similar to that in the single-impurity Kondo\n15 / 22\nproblem, in our case it applies to collective magnons in the ferromagnetic Kondo-\nHeisenberg lattice. Therefore, it involves more complex many-body effects than the\ntraditional single-impurity case. Further details are discussed in our response to\nComment 8.\nComment:\n- As plotted in Fig. 4, can the spin-flip processes induced by Kondo coupling occur in\nmaterials other than Fe - deficient Fe3 - xGeTe2 single crystals?\nResponse:\nWe thank the reviewer for this insightful question. We believe that the spin-flip\nprocesses induced by Kondo coupling are not unique to Fe GeTe . We expect that\n3-x 2\nsimilar phenomena can also occur in other systems where local moments exhibit long-\nrange magnetic order while interacting with itinerant electrons via Kondo coupling. In\nsuch systems, magnon damping provides a direct probe of the strength of Kondo\ninteractions---a central point of our study that we highlight and discuss in the\nconcluding section of the manuscript.\nComment:\n- The sentence in page 5 on \"...we consider Fe3−xGeTe2 demonstrates partial screening\nof local moments.\" How does the effective screening of magnetic moments specifically\nimpact the electronic properties of the system?\nResponse:\nWe thank the reviewer for this insightful question. The Kondo screening of local\nmoments in Fe GeTe enlarges the Fermi surface volume and enhances the effective\n3-x 2\nelectron mass, as reported by previous ARPES measurements (Ref. 29). This manifests\nin electronic transport as the slope change in resistivity around the incoherent-to-\ncoherent crossover. At the same time, the partial screening provides an additional decay\nchannel for spin excitations, directly impacting the magnon dynamics and contributing\nto the observed nonmonotonic temperature evolution of damping behavior. We have\ndiscussed this point in the section “FMKH model and spin-flip scattering mechanism”.\n16 / 22\n-------------------------------------------------------------------------------------------------------\nReviewer #3\n-------------------------------------------------------------------------------------------------------\nComment:\nThis manuscript detail an experimental study of spin waves in Fe deficient Fe3-xGeTe2\ncompound using inelastic neutron scattering. The experiments are complemented by\nnumerical simulations. The authors claim to have found anomalous increase in the\ndamping of the magnons at both high and low temperatures, displaying a minima at\nabout 90 K, and relate this to the Kondo coupling in magnetically ordered systems. I do\nfind this investigation of scientific interest, however, in my opinion, the findings in\nthemselves are not impactful enough to warrant publication in Nature Communications.\nI would instead suggest to pursue publication in a more specialized journal. In addition,\nI have some concerns about the data analysis and interpretation, as detailed below,\nwhich give me some doubt about the robustness of the presented (key) results.\nResponse:\nWe thank the reviewer for the assessment of our manuscript and for recognizing the\nscientific interest of our study. However, we respectfully but firmly disagree with the\nassessment that the findings lack the impact required for Nature Communications. Our\nwork constitutes the first demonstration that Kondo coupling in a long-range ordered\n3d-electron system, the metallic ferromagnet FGT, can be directly visualized by\nmeasuring magnon damping rates. We report the discovery of a logarithmic scaling law\nin magnon damping, which arises from the collective excitations of ordered local\nmoments, in contrast to the single spin fluctuations in intensively studied 4f-based\nheavy fermion systems. We attribute this novel scaling of magnon damping to a distinct\nmechanism driven by Kondo-coupling-induced magnon-electron scattering. Therefore,\nthe present study also provides a new approach for probing Kondo physics through\ncollective spin excitations.\nWe have also carefully addressed the reviewer’s concerns regarding data analysis and\ninterpretation. The additional clarifications and discussions included in the revised\nversion further strengthen the robustness of our key results. Detailed responses to these\nspecific points are provided below.\nComment:\nThe inelastic neutron scattering (INS) study is conducted using a cold-neutron triple-\naxis instrument. For the type of detailed analysis and interpretation presented in this\nmanuscript, I would definitely expect the authors to conduct a complete and thorough\nresolution analysis/deconvolution (resolution ellipsoid for a TAS instrument). This is\nespecially important for the Q-dependent data-set AND because the c-axis of this\nsample is so large (i.e. small c*). It is also vital to know the “vertical\nacceptance/resolution” for the instrument. Here any “contamination” from the\nresolution ellipsoid touching the base of any nuclear Bragg peaks can easily affect the\n17 / 22\nwidth of the magnon peak (i.e. the lifetime/damping). There is also no information\nabout the sample mosaic, i.e. how well aligned are the collection of samples (the\nmosaicity of the total sample used in the INS experiments). Some scans about this\nshould also be shown in the supplementary material in order to verify that the presented\neffect does not come from contamination of signal from nuclear Bragg or other\nmagnetic intensities.\nResponse:\nWe thank the reviewer for raising this important point regarding instrumental resolution\nand sample mosaic. In our analysis, the DHO fittings were already convolved with the\nsimulated resolution function, which is momentum- and energy-dependent. This\nfunction explicitly incorporates the measured sample mosaic (2.3° within the ab plane\nand 2.1° along the c axis) as well as relevant experimental settings such as collimation.\nNote that the elastic energy resolution is only about 0.2 meV, far smaller than the\nobserved magnon linewidths. After this convolution, we could reliably extract the\nintrinsic linewidths, which remain substantially broader than the instrumental\ncontribution. This demonstrates that the observed damping originates from intrinsic\nphysics rather than artifacts of resolution or mosaicity.\nFurthermore, the observed damping minimum is consistently observed at different\nmomenta for both the in-plane (Fig. 2e) and out-of-plane directions (Fig. 2f), and its\nmomentum dependence is also confirmed (Fig. 3). These results demonstrate that such\nmagnon damping behavior is intrinsic and not due to sample mosaic or contamination\nfrom nuclear Bragg scattering. In the revised manuscript, we have included the sample\nmosaic values and clarified the resolution convolution procedure.\nComment:\nI also have some questions regarding the analysis/results. In my understanding, the E0\nparameter that is plotted e.g. in Fig. 2(e,f), is simply the peak center for the magnon. In\nthis case, looking e.g. at the actual data of Fig. 2(c) and comparing it to results in Fig.\n2(e), it does not seem to match. For instance, the blue 4 K data shows a peak center\naround 0.3 meV in Fig. 2(c), however, in Fig. 2(e) the blue point at 4 K is around 1 meV.\nFurther, in Fig. 2(e) there is a blue point at Tc however, no red point is shown, but the\nred line indicate that there might be a point very high up (outside the graph). In fact,\nthe main point indicating that there is a strong increase in the damping for higher\ntemperatures is the 140 K point in Fig. 2(e). However, looking at the actual fits in Fig.\n2(c), for me it is not evident that this peak is much broader than the green 90 K data.\nFinally, the 4 K peak in Fig. 2(c) should according to plot in Fig. 2(e) be broader than\nall other peaks... it definitely does not look like this is the case (which also matches the\nfact that center E0 seems to be wrong). All these points makes me partly doubt the final\npresented (key) results about divergence of magnon damping at both high and low\ntemperature.\nResponse:\n18 / 22\nWe thank the reviewer for the careful examination of our data. The main source of\nconfusion seems to be the distinction between the fitted magnon energy E and the\n0\napparent peak center in the measured spectra. Since the magnons are strongly damped,\nwe fit the data using the damped harmonic oscillator (DHO) model, χ’’(Q, E) =\nγE E/[(E2-E 2)+(γE)2], where E is the magnon energy and γ is the damping rate.\n0 0 0\nImportantly, in this model the peak maximum does not necessarily coincide with E .\n0 0\nformula is approximated to a Lorentzian function. However, as γ increases, the spectral\nmaximum shifts progressively to lower energies. In the overdamped regime (γ/E >>1),\n0\nno true peak occurs near E , and only a broad hump appears at much lower energy. This\n0\nbehavior has been discussed in our previous work Ref. 31, including a comparison\nbetween E and the peak center (Fig. 5).\ndamped magnons than those at 45 K and 90 K. Consequently, the observed maxima are\nfar below the fitted E , explaining why the ~0.3 meV peak in the DHO fitting curve at\n0\n0\nfrom the DHO description of damped magnons.\nRegarding the specific point about T , the orange point representing γ overlaps with the\nC\nblue E point because of paramagnon contributions near T . These cannot be captured\nby the red line that diverges at T .\nC\nWe have revised the manuscript to clarify these points and explicitly explain the\ndistinction between E and the peak center to avoid confusion.\nFinally, I would have liked to see some information about the sample quality also in\nthis manuscript (and/or supplementary material). It would be of vital importance to\nshow that this Fe deficient sample is not phase separated. I also did not find any\ninformation about this in Reference 31 (which present data from the same sample).\nResponse:\nWe thank the reviewer for raising the question regarding possible phase separation in\nthe Fe-deficient sample. The single crystals used in this study were grown following\nthe widely adopted method reported in Phys. Rev. B 93, 014411 (Ref. 61 added), which\nhas demonstrated the stability of the grown phase. To our knowledge, no significant\nphase separation effect has been reported using this growth method. In that work, lattice\nparameters, Curie temperature, and saturation magnetization were shown to vary\nsmoothly with Fe concentration. Consistently, the single crystals grown in our work\nexhibit a single T ~160 K, as evidenced by both the magnetization data and the\nC\ntemperature evolution of the magnetic Bragg peak measured on the coaligned crystals\nused for the neutron experiments (Fig. 1 in Ref. 31). These results indicate a uniform\n19 / 22\nFe concentration across the investigated crystals. Furthermore, the inelastic neutron\nspectra in both Ref. 31 and the present study display well-defined spin excitations,\nproviding additional evidence against any macroscopic phase separation in our samples.\nWe note that local probes such as STM or STS can be sensitive to nanoscale variations\nin Fe content and may reveal spatially inhomogeneous responses. By contrast, our\nmeasurements, including macroscopic magnetization and inelastic neutron scattering\non coaligned crystals, probe the bulk and are inherently volume-averaged. Within this\ncontext, any minor local compositional fluctuations are averaged out, and their effects\nare essentially static and temperature independent, making them unlikely to influence\nthe observed nonmonotonic temperature dependence of magnon damping.\nIn the revised manuscript, we have added discussions on the sample quality and the\npotential influence of Fe vacancies during growth, clarifying their effects on the\nmagnetic properties.\nComment:\nIn summary, I cannot recommend publication of the current manuscript in Nature\nCommunications.\nResponse:\nWe respectfully acknowledge the reviewer’s overall assessment. We would like to\nemphasize that our study demonstrates, for the first time, a novel damping phenomenon\nof the collective excitations in a 3d-electron metallic ferromagnet, characterized by a\ndamping minimum and logarithmic scaling behavior. By combining momentum-\nresolved inelastic neutron scattering with the ferromagnetic Kondo-Heisenberg lattice\nmodel, we unveil the magnon damping and establish a new approach for probing Kondo\ncoupling through collective spin excitations. Fundamentally distinct from prior\nobservations in 4f-based systems, such magnon damping behavior is attributed to the\nKondo-coupling induced magnon-electron scattering. In the revised manuscript, we\nhave addressed the reviewer’s concerns regarding data analysis, interpretation, and\nsample quality, which we believe are now fully resolved. Taken together, we believe\nour results provide new insights of broad interest, and thus merit a publication in Nature\nCommunications.\n20 / 22\nSummary for the changes\n(1) Revise the abstract and the third paragraph to clearly distinguish our findings from\nthe conventional Kondo effect, which involves only isolated impurity spins.\n(2) In the section “Damping rate minimum and logarithmic scaling”, we explain why\noffset positions were chosen instead of the zone center (0, 0, 2) to study the\ntemperature evolution of magnon excitations.\n(3) In the same section, we emphasize that the DHO fittings were convoluted with the\nmomentum- and energy-dependent instrumental resolution.\n(4) We clarify that the magnon energy E obtained from DHO fittings does not\n0\nnecessarily coincide with the peak maximum in the measured spectra.\n(5) We have discussed the disorder or vacancy effects and clarify they cannot account\nfor the observed nonmonotonic temperature dependence of damping rate.\n(6) We also highlight how the quasi-two-dimensional van der Waals structure\ncontributes to the anisotropic magnon damping and softening for in-plane and out-\nof-plane directions.\n(7) In the section “FMKH model and spin-flip scattering mechanism”, we distinguish\nthe strong direct ferromagnetic exchange interactions in FGT from the typical\nantiferromagnetic RKKY interactions in f-electron systems.\n(8) We explain using a 1D FMKH model to capture the essential physics of the system.\n(9) We also clarify that the renormalization of the Kondo coupling in this work is\nformally similar to the single-impurity Kondo problem but, in fact, represents a\ncomplex many-body process.\n(10) In the section “Discussions”, we discuss the fundamental difference between the\nnonmonotonic temperature dependence of local 4f-moment relaxation rates\nreported in previous work and the damping minimum observed in our study. We\nhighlight the results in FGT reflect more complex many-body effects rather than the\nsingle-impurity scenario.\n(11) We further emphasize, based on field-theoretical analysis, that the FMKH model is\napplicable irrespective of dimensionality. In this context, the 1D FMKH model\n21 / 22\nserves as a minimal theoretical framework that successfully captures the essential\ndamping mechanism of a FM Kondo lattice.\n(12) In the section “Methods”, we add discussions on sample quality and the potential\ninfluence of Fe vacancies during growth.\n(13) We describe the use or removal of a cooled Be filter in different experimental runs.\n(14) We also add information about the sample mosaic of the coaligned crystals in the\n“Methods” section.\n(15) We add additional references Refs. 56-59, 61.\n22 / 22"}, {"round": 2, "reviewer_comments": {"reviewer_1": "I am satisfied with the revision. The authors have clarified the points I raised and demonstrated a important advance compared to the early reports on the heavy fermion compounds. I am happy to recommend this manuscript for publication in Nature Communications.", "reviewer_2": "I appreciate the authors have carefully answered my questions and revised the manuscript correspondingly. Now I can recommend its publication in Nature Communications.", "reviewer_3": "I acknowledge that the authors have undertaken a substantial revision of their manuscript and have addressed most of the comments raised by myself and the other two reviewers. However, two of my major concerns remain insufficiently resolved:\n1. Novelty and significance:\nThe authors still do not convincingly demonstrate that their work represents a substantive advance beyond existing studies. Both Reviewer #1 and I previously raised this issue, and the current revision does not provide new evidence or arguments to establish the work as more than an incremental contribution.\n2.Sample quality:\nMy concerns regarding the quality of the samples remain unaddressed. The authors have not presented additional data or analysis to substantiate the quality of their samples. The authors' statement that:\n“The single crystals used in this study were grown following the widely adopted method reported in Phys. Rev. B 93, 014411 (Ref. 61 added), which has demonstrated the stability of the grown phase.”\nis insufficient as supporting evidence. To convincingly establish sample quality, particularly for a sample mosaic, the authors should provide direct experimental evidence or characterization data confirming the integrity and homogeneity of their samples.\nIn summary, given these unresolved issues, I find that the revised manuscript still falls short of the standards required for publication in Nature Communications. I therefore cannot recommend acceptance of the manuscript in its current form."}, "author_response": "Response to the Reviewers\nWe sincerely thank all three reviewers for their time, careful reading, and constructive\nevaluations of our revised manuscript. We are very pleased that both Reviewer 1 and\nReviewer 2 have found our responses satisfactory and have recommended the\nmanuscript for publication in Nature Communications. We greatly appreciate their\nrecognition of the significance of our findings and the quality of our experimental work.\nReviewer 3 acknowledged that most of the earlier comments had been properly\naddressed but retained concerns regarding the novelty of our work and the quality of\nthe samples. We have carefully revisited these points and further clarified both aspects.\nWe respectfully maintain our original interpretation and manuscript text concerning the\nnovelty, as we believe they already convey the significance and physical implications\nof our results clearly and accurately. In addition, we now provide new experimental\nevidence confirming the integrity and homogeneity of the samples through rocking-\ncurve scans of the whole single-crystal assembly and magnetization measurements\nfrom multiple growth batches.\nWe value the constructive feedback from all reviewers, and we have addressed all\nremaining technical points in the revised manuscript. The changes are marked in red,\nand a summary of all modifications is provided after the point-by-point responses. We\nbelieve that the current revision fully resolves all technical and conceptual issues raised\nby the reviewers and that the manuscript now meets the standards of Nature\nCommunications in both significance and broad interest.\n1 / 6\n-------------------------------------------------------------------------------------------------------\nReviewer #1\n-------------------------------------------------------------------------------------------------------\nComment:\nI am satisfied with the revision. The authors have clarified the points I raised and\ndemonstrated a important advance compared to the early reports on the heavy fermion\ncompounds. I am happy to recommend this manuscript for publication in Nature\nCommunications.\nResponse:\nWe thank the reviewer for the positive evaluation and the recommendation for\npublication. We are very pleased that the revisions have clarified our key findings and\ntheir significance.\n-------------------------------------------------------------------------------------------------------\nReviewer #2\n-------------------------------------------------------------------------------------------------------\nComment:\nI appreciate the authors have carefully answered my questions and revised the\nmanuscript correspondingly. Now I can recommend its publication in Nature\nCommunications.\nResponse:\nWe thank the reviewer for the encouraging comments and recommendation for\npublication. We appreciate the reviewer’s recognition of our careful revision and the\nscientific value of our results.\n2 / 6\n-------------------------------------------------------------------------------------------------------\nReviewer #3\n-------------------------------------------------------------------------------------------------------\nComment:\nI acknowledge that the authors have undertaken a substantial revision of their manuscript and have addressed most of the comments raised by myself and the other two reviewers. However, two of my major concerns remain insufficiently resolved:\nResponse:\nWe thank the reviewer for the feedback. We are pleased that most of the earlier concerns\nhave been resolved. Below we address the remaining points regarding novelty and\nsample quality in detail.\n1. Novelty and significance:\nThe authors still do not convincingly demonstrate that their work represents a substantive advance beyond existing studies. Both Reviewer #1 and I previously raised this issue, and the current revision does not provide new evidence or arguments to establish the work as more than an incremental contribution.\nResponse:\nWe thank the reviewer for the comment. As detailed in our previous response, our work\nuncovers a novel phenomenon of Kondo coupling. Specifically, we demonstrate that\nthe low-temperature logarithmic scaling of magnon damping arises from Kondo-\ncoupling induced magnon-electron scattering in a metallic quantum ferromagnet. This\nfinding establishes a new framework for probing Kondo coupling through the dynamics\nof ordered local moments. This is fundamentally distinct from the conventional Kondo\neffect involving isolated impurity spins, including the cases mentioned by Reviewer 1\nin the first round of review. Reviewer 1 has recognized this distinction and fully\nappreciated the novelty of the work.\nWe therefore respectfully maintain our original interpretation and manuscript text,\nas we believe they already convey the novelty and physical implications of our results\nclearly and accurately.\n2.Sample quality:\nMy concerns regarding the quality of the samples remain unaddressed. The authors have not presented additional data or analysis to substantiate the quality of their samples. The authors' statement that:\n“The single crystals used in this study were grown following the widely adopted method reported in Phys. Rev. B 93, 014411 (Ref. 61 added), which has demonstrated the stability of the grown phase.”\n3 / 6\nis insufficient as supporting evidence. To convincingly establish sample quality, particularly for a sample mosaic, the authors should provide direct experimental evidence or characterization data confirming the integrity and homogeneity of their samples.\nResponse:\nWe thank the reviewer for pointing out the need for more evidence regarding the sample\nquality. In the revised manuscript, we have addressed this concern by including new\nexperimental data to confirm the integrity and homogeneity of the single crystals used\nin our study.\nFig. R1. a, Rocking curves measured at the (1, 0, 0) and (0, 0, 6) Bragg reflections for the\ncoaligned crystal array. b, Magnetization curves of four single crystals from different growth\nbatches, showing consistent magnetic behavior.\nSpecifically, we have included rocking curve measurements conducted on the entire\nsample array composed of coaligned single crystals, which were grown in different\nbatches using the method described in Ref. 61. The data show that the coaligned crystals\nexhibit a uniform mosaic structure, with ~1.6° spread in the a-b plane and a ~1.7° spread\nalong the c axis, as shown in Fig. R1a.\nIn addition, magnetization measurements were carried out on randomly selected single\ncrystals from four different growth batches, as shown in Fig. R1b. The results\ndemonstrate that these crystals, regardless of batch, exhibit nearly identical magnetic\nsusceptibility curves, with a consistent Curie temperature T ~160 K. This is in good\nC\nagreement with our prior magnetization and resistivity measurements, as well as with\nthe temperature evolution of the magnetic Bragg peak observed in the same coaligned\ncrystals (Ref. 31).\nThese findings confirm the integrity and homogeneity of the single crystals we used.\nWe have added Fig. R1 to the Supplementary Materials and have discussed it\naccordingly in the revised manuscript.\n4 / 6\nIn summary, given these unresolved issues, I find that the revised manuscript still falls short of the standards required for publication in Nature Communications. I therefore cannot recommend acceptance of the manuscript in its current form.\nResponse:\nWe thank the reviewer for the feedback and evaluation of our revised manuscript. As\ndiscussed extensively in our previous and this response, the novelty of our work lies in\nthat we uncover a new scaling law of magnon damping of collective spin waves, and\nidentify its underlying physical mechanism as being induced by Kondo coupling. This\nmechanism is fundamentally different from the conventional Kondo effects observed\nin impurity-spin systems. This has been recognized by Reviewer 2, and now by\nReviewer 1 also. We respectfully maintain that the revisions made have thoroughly\naddressed the reviewer’s concerns. Concerning the sample quality, we have added\nrocking curve measurements on coaligned single crystals and magnetization results\nfrom multiple growth batches, which provide compelling evidence for the homogeneity\nand integrity of our samples.\nWith these, we humbly believe that the work is publishable as it is.\n5 / 6\nSummary for the changes\n(1) Revise the Methods section to clarify the homogeneity of the single crystals and the\nmosaic spread of the coaligned sample array used for the neutron measurements.\n(2) Add a new supplementary figure (Supplementary Fig. 5), showing rocking-curve\nand magnetization data that verify the integrity and batch-to-batch consistency of\nthe crystals.\n6 / 6"}, {"round": 3, "reviewer_comments": {"reviewer_1": "I believe the sample quality concern has been satisfactorily addressed."}, "author_response": ""}]

Decision letter