Non-perturbative intertwining between spin and charge correlations: A "smoking gun" single-boson-exchange result

We thank the Referee for the careful review of our manuscript, for her/his overall positive evaluation, as well as for her/his detailed observations.

Below, we detail our Reply to all specific points raised in her/his report:

The manuscript presents a very detailed study and is written in a clear manner. However, before I can recommend this work for publication, I would like the authors to address to the following questions:

My main question concerns the interpretation of the results obtained. The authors relate the transition from the high- to intermediate-temperature regime, and consequently the appearance of the specified frequency structure of the generalised charge susceptibility, to the formation of the local magnetic moment. From my point of view this relation is not very well explained and is not well justified.

If this transition is identified by the divergence of the irreducible vertex function in the charge channel, it can only be related to the breakdown of the many-electron perturbation expansion following Refs. [1,3,7,9,10,13-16,18,19] cited by the authors. In Ref. [22], the divergence of the vertex function was associated with the formation of the local magnetic moment, as in the low-temperature and strong-coupling regime the "divergence curve" aligned with the Kondo temperature. However, there was no justification that the divergence of the vertex function could be connected to the formation/destruction of the local magnetic moment in the high-temperature regime.

If the transition is identified by (quoting the authors) "a relative flat (Curie) behavior of the quantity $T \chi^{sp}_{\omega=0}(T)$,” then this condition is imprecise. It would be helpful if the authors show the results for the local susceptibility and explain how they identified the transition point. In fact, the deviation from the Curie behavior of the spin susceptibility is rather smooth (see, e.g., [PRB 99, 165134 (2019)]), which usually does not allow one to accurately pinpoint the formation of the local magnetic moment (see, e.g., [arXiv:2112.02881]). In addition, in [PRB 105, 155151 (2022)] it was argued that the formation of the local magnetic moment cannot be captured by the behavior of the static spin susceptibility, because (quoting the authors of that work) the spin susceptibility "cannot distinguish the fluctuations of the local magnetic moment from the spin fluctuations of the itinerant electrons that also contribute to the susceptibility, especially in the paramagnetic regime."

We thank the Referee for this comment. Indeed, the study of the relation between the spin- and the charge-sector in the different regimes (and especially of the local moment one) is one of the central points of our work. Hence, it is important that this aspect is presented in the most clear and convincing way in our reply and in the revised manuscript.

Let us start by stressing (what we have also done in the revised text) that it is not our aim, in this work, to introduce/define or even improve criteria for delimiting the different physical regimes. Indeed, as the Referee points out, this task would be quite hard (if not impossible from a purely rigorous perspective), considering that the different regimes studied in our selected models (HA, AIM, as well as the paramagnetic DMFT solution of the HM on the left side of the MIT) are separated by crossover regions and not by sharp phase transition lines. Our goal is, instead, to precisely rationalize the mechanisms controlling, on the two-particle level, the physics of charge localization, which arguably is "the other side of the coin'' of the local moment formation. In particular, we aim at eventually clarifying how the specific way in which the charge localization gets encoded in the corresponding generalized susceptibilities is linked to the physics underlying the local moment formation and (where applicable) its Kondo screening. In fact (cf.~Introduction and Sec.~II), it was noted in previous works, on an empirical basis, that the freezing of the on-site charge response in the strongly correlated regimes of several basic models was associated to a marked suppression of the diagonal entries of the (corresponding) generalized susceptibility $\chi^{\nu \nu'}$ (which could become quickly negative at low-frequencies) and to a simultaneous slight increase of the off-diagonal elements. However, the essentially empirical nature of such observations prevented to draw rigorous conclusions, leaving the question open (which was posed to some of us several times in conferences and discussions) why the reduction of the on-site charge response driven by correlations was occurring in this precise way rather than, e.g., through an uniform suppression of all matrix elements of $\chi^{\nu \nu'}$, or with a larger suppression of the off-diagonal ones, etc. As the Referee rightly mentioned, this question is also tightly linked to problem of the breakdown of the self-consistent perturbation theory, since the specific (abovementioned) way in which the suppression of on-site charge fluctuations takes place is primarily responsible for the multiple sign-flips of the eigenvalues of $\chi^{\nu \nu'}$, and hence for all the related consequences (divergences of irreducible vertices, crossings of different solutions of the Luttinger-Ward functional, convergence to unphysical results of the many-electron expansion, etc.). Evidently, if the freezing of the on-site charge fluctuations had occurred in a qualitatively different fashion on the two-particle level than the way described before, it might have been possible for self-consistent perturbative approaches to capture the local moment regime physics (including the associated Mott insulating phase in DMFT). We think that our diagrammatic decomposition provides a clear-cut answer to the question above. The obtained results precisely identify the scattering processes (i) mostly responsible for the progressive suppression of the low/inter-mediate diagonal frequency entries of $\chi^{\nu \nu'}$, i.e.~those associated to the electronic scattering with a single spin mode, (ii) as well as those causing the slight overall enhancement of the off-diagonal terms, due to multiple scattering with collective bosonic excitations. It is important to stress that (i) explains then, in a perfectly natural way, why the freezing of on-site charge fluctuation due to the local moment formation happens in the specific way we observe it: The more well-defined the local magnetic moment will be, the longer will be its lifetime (Lifetime, which becomes infinite in the limiting case of the HA, to be regarded, in this sense, as the "perfect gas'' analog for the local moment physics.). In (Matsubara) Fourier space this trend gets immediately reflected in a progressive frequency-localization of the suppressive effects originated by the corresponding single spin-exchange processes on the diagonal entries of $\chi^{\nu \nu'}$. At the same time, this lifetime effect, though crucial, would have not been enough alone to allow for a correct transfer of information between the different channels in the local moment regime. The latter is made possible by the simultaneous low-frequency enhancement of the corresponding spin-fermion scattering amplitude (i.e., of the so-called triangular vertex) w.r.t.~its perturbative/asymptotic value. This diagrammatic identification, which appears numerically quite solid in the three model considered, provides an clear-cut explanations of the question why the freezing of the on-site charge fluctuations happens in the specific nonperturbative manner observed, and how the relevant information (enhanced on-site magnetic response, suppressed on-site charge response) gets transferred between the different physical sectors. In this perspective, in the low-temperature limit of the HA (where the physics of the local moment is essentially perfect, up to vanishingly small exponential corrections of order $\sim e^{- \beta U}$), by estimating the minimal magnitude of the spin-fermion vertex to observe a sign-flip on the diagonal entries of $\chi^{\nu \nu'}$, we were able, finally, to clarify why the size of the frequency-region $[-\nu_{max}, \nu_{\max}]$ where (nonperturbative) negative diagonal values of $\chi^{\nu \nu'}$ are observed scales precisely as $\nu_{max} = \frac{\sqrt{3}}{2} U$. Indeed, the previously empirically determined scaling factor of $\frac{\sqrt{3}}{2}$ finds its most natural explanation in the prefactor of the single spin-exchange contribution, further supporting the validity of our analysis. As we will detail better below, by revising our manuscript we tried to better emphasize the main goal of our study as well as the relevance of the results obtained in this perspective, and to refine/modify imprecise (and, eventually, intrinsically non-conclusive) statements about the borders of the different regimes.

On the contrary, the transition between the low- and intermediate--temperature regimes is clearly defined by the Kondo temperature. Therefore, it would be helpful if the authors specify the values of the Kondo temperature for the systems under consideration and provide an explanation of how they were obtained, as the definition of the Kondo temperature for these systems is not unique. Actually, the issue regarding the formation/destruction of the local magnetic moment is even less clear for the Hubbard atom, which does not have a Kondo regime.

Following the suggestion of the Referee, in the revised manuscript, we have now reported the estimated value of the Kondo temperature ($T_K$) for the AIM considered, as well as of the "effective'' $T_K$ for the DMFT solution of the Hubbard model. The former has been extracted by the temperature dependence of the local magnetic susceptibility $T \chi_m(T)$, namely by matching it to the universal temperature behavior of the Kondo problem, following the procedure detailed in the Appendix A of Phys.~Rev.~B 97 245136 (2018) as well as Sec.~II in Supplemental Material of Phys.~Rev.~Lett. 126, 056403 (2021). We note that this procedure yields, at the $U$ considered, even on a quantitative level the corresponding textbook wide-band limit value for $T_K$ [see, Eq.~(6.109) and ff.~at p.~165–166 of Chap. 6.7 in A.~C.~Hewson, "The Kondo Problem to Heavy Fermions'' (Cambridge University Press, Cambridge, 1993)] (As $T_K$ marks a crossover scale, as the Referee also mentioned, other definitions/criteria could have been chosen. For instance, by estimating $T_K$ via the width of the corresponding Kondo-peak in the spectral function one typically gets larger estimates (up to five times!) than those obtained from the temperature dependence of $T \chi_m(T)$. The latter criterion represents, however, one of the most common choices made in the literature.) , except for the (marginally small!) corrections due to the finite (albeit large) bandwidth of the bath electrons. The precise value of $T_K$ in the energy units of our AIM (Namely, an AIM with a constant DOS of the bath electrons in the interval $[-D,D]$ with $D=10$, hybridization amplitude $V=2$ and impurity on-site interaction $U=5.75$) reads $T_K \simeq \frac{1}{65} \simeq 0.015$. As discussed Phys.~Rev.~Lett. 126, 056403 (2021), in the strong-coupling regime (which applies to the value of $U$ considered here) this (textbook) value of $T_K$ is extremely well approximated by a specific condition on the lowest Matsubara entries of the generalized charge susceptibility, namely that $\chi_c^{\pi T, \pi T} = \chi_c^{\pi T, -\pi T}$. As for the DMFT calculations, where the AIM plays an auxiliary role for the self-consistent determination the corresponding dynamical mean field, the procedure described above cannot be straightforwardly followed, as the auxiliary AIM itself (as well as its Kondo temperature) depends (for a fixed $U$) on the temperature itself. Here, however, by resorting to the criterion based on the lowest Matsubara-frequency mentioned above, one could determine the temperature at which the effective $T_K$ of the corresponding auxiliary AIM is crossed, i.e.~$T_K(T) = T$. For our DMFT calculations on the Bethe lattice, with $U=2.2$ in unit of the half-bandwidth, its estimation yields $T_K \simeq \frac{1}{50}$. Finally, in the Hubbard Atom the local spin operator (as well the local charge operator) is a constant of motion of the problem, allowing to regard this system as an ``ideal realization'' of the local moment physics. The absence of any Kondo screening, yields perfect local moment features in the low-temperature limit, with negligibly small $e^{-\beta U}$ corrections. In that case, the local moment physics can be only be degradated by the thermal activation of the excited states (with $0$ and $2$ electrons, respectively), which occurs at temperatures of the order of the corresponding energy gap, i.e. $T \sim \frac{U}{2}$. Indeed, a brief glance on the temperature dependence of the susceptibilties, allows to appreciate how, in the case of the HA, both the fullfillment of the Curie behavior of the spin response, as well the corresponding exponential suppression of the charge response, become virtually perfect already at (or below) $T \sim 0.5 < \frac{U}{2}$.

To summarise, it would be helpful if the authors: 1) Provide clear specifications for the transition points between the high-, intermediate-, and low-temperature regimes and elaborate on how these temperatures were calculated. 2) Justify the relation of the mentioned transitions to the formation/destruction of the local magnetic moment, or alternatively, refrain from making such a connection. 3) Demonstrate that the change in the frequency structure is indeed happens at the transition point and not somewhere else in the phase.

Considering the questions raised by the Referee, and consistent with our reply, in our revised manuscript we have now refined the presentation of the aims of the paper, underlying that our main goal is not to provide univocal definitions for the crossover borders between the different regimes (which would be, {\sl a priori}, an unfeasible task, due the lack of sharp phase-transitions in the cases considered), but rather to unambiguously identify the scattering processes responsible for the correct intertwining between different physical sectors, i.e. the processes allowing the large local magnetic response due to electronic localization to be accompained (as it should !) by a corresponding suppression of on-site charge fluctuations. We have also emphasized better in the revised version, how this result has allowed us, eventually, to clarify, why such interplay happens in the way we observe it, relating its manifestations to the intrinsic properties of the local moment formation (such as, e.g., to its characteristic long lifetime). In the revised paper, we also provide a concise justification for the choice of parameters (essentially, as $U$ is kept fixed, for the three selected temperatures), we made to study the physics of the different regimes. In particular, for the HA, we've chosen $T = 2 ~ \frac{U}{2} (\beta =0.5)$ for illustrating the behavior of the perturbative regime (of course here, also higher temperatures, but this would have further reduced the "resolution" of our Matsubara susceptibility matrices), and $T=0.1 << \frac U2 (\beta=10)$ for describing the local moment regime. For the AIM (for which we used the same interaction value $U$ as in the HA), the choice of the first two ''higher" temperatures, representative of the perturbative and of the local moment regime is the same as above (as are both much larger than $T_K$), while for the screened (Kondo) regime we selected $T = \frac{1}{60} \sim T_K \simeq \frac{1}{65}$ (compare Fig.~1 of the reply to referee 1). Of course, going along the line of thoughts of the Referee, one might indeed try to exploit the sharper sign-structures characterizing in the generalized charge susceptibility to define new/complementary criteria for delimiting the different regimes of the crossover, similarly to what was done in Phys.~Rev.~Lett. 126, 056403 (2021) for the Kondo Temperature. For instance, for the high-$T$ border $T^{*}(U)$ of the local moment region, one could use the sign-flip from positive to negative of the lowest eigenvalue of the generalized susceptibility, which indeed for large coupling display a scaling with $T^* = \frac{\sqrt{3}}{2}U$. The introduction of such sharp criteria in the charge sector might be even more useful out-of-half filling, where the temperature features in the spin-sector might become even more elusive. At the same time, the introduction of any of such crossover criterion represents (intrinsically) an arbitrary choice, hence, not being the main goal of our study, we prefer not to address explicitly this issue in this study.

In addition, I have two small questions: 1) Could the authors comment on why the charge susceptibility was chosen to study the effect of the formation of the local magnetic moment? If this effect ``originates from the electronic scattering on the spin susceptibility," can it be observed directly by examining the spin susceptibility?

This represents an important point, indeed. On the one hand, as discussed also above, the local moment formation and the associated freezing of local charge fluctuations are the two sides of the same coin ("simul stabunt, simul cadent'', i.e. one cannot have one effect, without the other). Even, the strong intertwining between the two scattering channels (mediated by the enhanced value of the triangular vertex in the local moment regime) is eventually responsible for the breakdown of the perturbation expansion and all its related manifestations. Hence, it would be reasonable to search for the presence of characteristic features of the local moment formation in the generalized magnetic susceptibility, too. On the other hand, the two channels are strongly intertwined, though, they are certainly not equivalent, since the on-site spin response is enhanced and the charge response is frozen. This difference is largely reflected in the corresponding generalized susceptibilities. In particular, one observes that the generalized spin susceptibility displays an enhancement at all frequencies in the local moment regime, whereas the low-temperature Curie behavior of the susceptibility would be associated to a rather featureless positive structure extended on all fermionic Matsubara frequencies $|\nu|,|\nu'| \leq U$. This overall strong, but rather diffuse enhancement makes the fingerprints of the local moment formations, in same sense, not so easy to be directly "read'' from the overall structure of the generalized magnetic susceptibility. On the contrary, the suppression effects on the generalized charge susceptibility (associated to the different signs of its $\uparrow \uparrow$ and $\uparrow \downarrow$ counterparts) is reflected in sharp frequency structures of {\sl different signs} in the charge sector, which are very easy to identify, even at a first glance, and to be directly compared to those observed in other regimes. Beyond this practical reason (a much natural identification procedure), it is also worth to stress here a more general point: The freezing of the on-site charge fluctuations is a crucial aspect in strongly correlated electronic models: It plays an essential role in controlling the electronic mobility properties of the systems. In this respect, we note that precisely these nonperturbative suppressive effects in the charge sector, associated to the formation of the local moments, are responsible for the occurrence of Mott-Hubbard metal-insulator transition in DMFT. Indeed, as mentioned above as well as in Phys.~Rev.~Lett. 126, 056403 (2021), within (even quite advanced) self-consistent perturbation approach, such as truncated fRG and the parquet approximation, even in the presence of local moment features in the spin sector, the freezing of the on-site charge fluctuations does not take place, due to a too little intertwining between the two channels. These considerations further support the choice of focusing on the generalized charge susceptibility.

2) Could the authors specify the parameters used to obtain the results shown in Fig. 1?

We now remarked the parameters in the figure caption.

Please correct two typos: 1) Page 3 - “(s. below)” 2) Page 5 - “In order to so

We thank the referee for the remarks and have corrected the typos.

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We thank the Referee for carefully reading our manuscript, for the positive evaluation of our work and her/his constructive observations.

The Referee has asked us to consider specific points to be addressed prior to publications. We have considered all of them very thoroughly and included the corresponding changes into the manuscript text, figures and the appendix. Specifically, we report below a detailed reply to all the observations of the report. The main questions posed by the Referee are the following ones:

The central statement of the paper is that the spin channel provides the most important contribution to the frequency dependence of charge susceptibility, and capable to describe various features of charge susceptibility observed previously in Ref. [22]. To clarify the relative role of vertex corrections and the susceptibility itself, it would be helpful to see in Figs. 3,5,8 the spin contribution with $\lambda_{spin}=1$.

In order to better clarify the role of the vertex correction to the susceptibility in a coherent way w.r.t.~the flow of the paper, we have included a comparison of the spin contribution to the generalized charge susceptibility obtained with and without approximating $\lambda_{\text{sp}}=1$ in Fig.~12 in Sec.~3.4 and added a corresponding explanation in the text. Panels 2-4 of this figure clearly show the importance of $\lambda_{\text{sp}}$ in the local moment and Kondo regime of the AIM and the HA. The relative role in the local moment regime is an enhancement of the absolute value with respect to the non-interacting limit. This situation is reversed in the Kondo regime, where the spin contribution is largely suppressed in absolute value due to the screening effect of the electronic bath. As described in the text here the Hedin spin-vertex has values smaller than $1$ i.e. its non-interacting limit. For the DMFT solutions in the corresponding two regimes (not shown) the very same considerations apply. To provide a full frequency picture of the effects of this approximation, we have also added the new Appendix C, where a colorplot showing the whole frequency structure (including the frequency off-diagonal elements of the SBE spin contribution) is reported and briefly discussed.

It is not fully clear what is shown by dashed red lines (which are explained as the contribution proportional to $\chi_s$, but not explained in more details).

Regarding the explanation of the red dotted line (note that it was incorrectly indicated as "red dashed" in the main text, which we have corrected in the revised manuscript) in Fig.~3, 5 and 8, we agree that it needs to be extended. Specifically, the spin contribution of Eq.~(11) can be formally split into two parts by inserting Eq.~(10) i.e.

(note that $\nu'-\nu$ is a bosonic frequency). Keeping only the latter term one gets which is precisely the one we indicated with the red dotted line. This specific SBE contribution, being directly proportional to $\chi^{\text{sp}}(\omega)$ is of most interest for its transparent link to the physical spin response. In order to better clarify this point, we have now added its explicit expression in the caption of Fig.~3 and the main text.

It might be also useful if the authors provide plots of $\chi_s(T)$ and/or $T*\chi_s(T)$ (possibly in Appendix) to understand evolution of the susceptibilities in different regimes.

We agree that such a plot would be useful and have indeed added it together with the new Appendix D. It is also reproduced in the attached structured version of this reply for clarity.

Is it also a coincidence that the behaviour of different channels in high-temperature and Kondo regime is somewhat similar (although it is different by magnitude)?

We assume that the question is mostly referring to the comparison of the data plotted in Fig.~3 and Fig.~8, because only there we show the separate contributions of all channels to the diagonal frequency entries of the generalized charge susceptibilities in the perturbative and in the Kondo screened regime. In this case, we should first emphasize that --on a general level-- one expects that all contributions of our SBE decomposition display larger intensities for lower frequencies along the diagonal, due to their asymptotic decay at high frequency. Further, their specific signs (positive for the bubble, negative for the spin contribution, etc.) appear to be fixed at half-filling, where special particle-hole symmetric properties hold (such as, e.g., that the on-site Matsubara Green's function is purely imaginary). Hence, to a first glance, the structures of the different contributions along the diagonals (unless they are not completely suppressed) might look qualitatively similar. It is also true, however, that beyond this general observation, additional similarities can be noted between the low-frequency perturbative and Kondo regime, due to the screening effects active in the latter case. These are responsible, for instance, of the low-frequency increase (w.r.t.~to the local moment regime) of the bubble term as well as of a moderation of the suppressive contribution of the spin channel, which both drive the (relative) low-$T$ revival of on-site charge response. Obviously, the similarity is not complete. By looking at a more quantitative level, differences also emerge, such as the much smaller/larger contribution of the singlet channel/$U_{\rm irr}$ in the Kondo w.r.t.~to the nonperturbative regime, as well as the almost perfect identification of the spin contribution with its component proportional to the physical susceptibility in the Kondo regime. Eventually, even more evident differences between the perturbative and the Kondo regime can be observed when comparing the off-diagonal frequency structures (e.g., by comparing the third column panels of Fig.~4 and Fig.~9, where the corresponding multiboson contribution is shown).

The title of the paper "Non-perturbative intertwining..." looks to me somewhat misleading. Indeed, the authors consider purely perturbative contributions to the charge channel (apart from the irreducible one, which does not play big role in their results). All the non-perturbative information is therefore hidden in the triangular spin vertex and spin susceptibility, which behaviour the authors almost do not analyze. I suggest the authors also to extend the discussion on the non-perturbative aspects in Conclusion and text of the paper.

This question is of high importance for our work and certainly requires additional clarification (both in the reply and the revised text), as it also touches relevant aspects, which have emerged during the presentation of our results to other colleagues in informal discussions and conferences. Indeed, the Referee is quite right in noticing that one of the pivotal effect we described in the paper, i.e. the sign flip of the diagonal elements of the generalized charge susceptibility is driven by the SBE (and two-particle) reducible scattering processes in the spin channel. In the SBE decomposition, however, no perturbative assumption is -a priori- made, and, as the Referee also noted, the two main constituents of the spin SBE-contribution clearly identified as responsible for the systematic suppression of the diagonal entries of $\tilde{\chi}_c^{\, \nu \nu'}$, namely (i) the (static) physical spin response and (ii) the triangular spin-fermion vertex are the exact ones (for the corresponding case considered) without any a priori restriction to any perturbative approximation. It is important to emphasize, here, that precisely this clear-cut identification via SBE decomposition, which was missing in previous studies (including ours), allows to unveil the physics underlying the breakdown of the self-consistent many-electron perturbation-expansion. In particular, in previous studies, it was just noticed, essentially on a mere empirical basis, that in several fundamental models for strongly correlation, the suppression of on-site charge response occurring the local moment regime of the corresponding phase-diagrams was mostly driven by a strong suppression of the lowest frequencies diagonal entries of the generalized charge susceptibility and, not, e.g., by a generic/uniform reduction of all its matrix elements (which would have been also possible\footnote{For instance this may indeed happen, in the case of a reduction of the density of states of the non-interacting Hamiltonian}. This specific feature is the one determining the breakdown of the self-consistent perturbation expansion, as the suppressed (and then even negative) diagonal entries of $\tilde{\chi_c^{\nu \nu'}}$ causes a sign-flip of its eigenvalues, and, hence, whenever one eigenvalue vanishes, the associated divergences of the irreducible vertex function, the non-invertibility of the corresponding Bethe-Salpeter equation (BSE), and the crossing to physical and unphysical solutions in the Luttinger-Ward functional formalism. A legitimate question posed by many colleagues (as well as by ourself) was then to understand whether the suppression of the on-site charge response associated to a local moment should necessarily occur in this precise fashion (which then unavoidably leads the perturbative breakdown), and, if yes, why this is the case. The identification of the (overall negative!) spin-SBE contribution to $\tilde{\chi}_c^{\, \nu \nu'}$ as the main suppression mechanism of the on-site charge response, presented in this manuscript, has finally provided a clear-cut answer to these questions, in terms of the two main ingredients of the spin-SBE scattering processes mentioned above. Specifically, in the local moment regime (i) the long life-time (actually even infinite in the perfect realization of the local moment, i.e. the Hubbard Atom) of the on-site spin correlations is directly reflected in a selection rule of the major suppression effects of the local charge-fluctuations for $\nu ~ \nu'$ (whereas $\nu \equiv \nu'$ in the "perfect" HA case, where the local spin is a conserved quantity) (ii) the spin-fermion coupling (triangular vertex) gets enhanced w.r.t. its perturbative value of $1$ at low-fermionic frequencies $\nu$. Evidently, the combination of (i) + (ii) explains why the suppression of the on-site charge response, which is unavoidably associated to the formation of a local moment must occur in the precise way observed in the previous work, leading necessarily to a divergence of the irreducible vertex, and, hence, to the breakdown of the self-consistent perturbation expansion. Our analysis, thus, rigorously clarifies the physical nature of the perturbation theory breakdown in all fundamental models considered: The simultaneous enhancement of the on-site magnetic static response and suppression of the on-site charge one, which are both, indeed, intrinsic features of the local moment physics. Hence, any (self-consistent) perturbative approach is bound to fail in describing a proper suppression of the charge fluctuations in the presence of a local magnetic moment, due to the intrinsic impossibility in self-consistent perturbation theory of flipping the sign of any of the eigenvalues of $\tilde{\chi}_c^{\, \nu \nu'}$ , which will remain all positive, as in the corresponding non-interacting case of the model considered. This specific (but relevant!) drawback of self-consistent perturbation approaches has been explicitly observed, e.g. in (truncated) functional renormalization group (fRG) and parquet approximation (PA) calculations, where the local charge response was found to monotonically increase when reducing the temperature even in the local moment regime, reflecting the too weak suppression of $\tilde{\chi}_c^{\, \nu \nu'}$ for $\nu \sim \nu'$ (indeed the diagonal elements of $\tilde{\chi}_c^{\, \nu \nu'}$ remain positive in all fRG and PA dataset). This way, one can eventually understand that the breakdown of the perturbative description in Hubbard model systems is intrinsically rooted into the strong communication between the different physical sectors (magnetic vs. charge, but also particle-particle/pairing), which is essential to yield a self-consistently coherent picture of the local moment physics in its entirety, where the enhancement of the static local spin response must consistently occur together with the suppression on-site charge (and pairing) fluctuations (Note that evidently the same consideration will apply, mutatis mutandis to the formation of local pairs in the case of an attractive on-site interaction (negative $U$)). We note in passing that this strong interplay between the different sectors also represents a crucial ingredient for the (indeed nonperturbative in $U$!) dynamical mean-field theory description of Mott metal-insulator transitions.

We note here -although this is beyond the scope of the present work- that, consistent with our considerations, the unphysical solutions obtained in bold (=self-consistent) diagrammatic Monte Carlo after crossing the first vertex divergence line (i.e. in the nonperturbative regime) are precisely characterized by an unphysical metallicity even in the local moment regime, with a too large charge mobility and even a value of double-occupancy increasing with $U$ (According to several studies bold diagrammatic Monte Carlo resummations do converge also in the nonperturbative regime, albeit not to the correct/physical solution: this is referred to as "misleading convergence" of the self-consistent perturbation expansion, which appears after crossing the first vertex divergence line.).

Another point: In the beginning of Sect. 3.2 the authors mention formation of relatively flat part of $T\chi_s(T)$ and refer to Refs. [38,49]. However, these references refer to Anderson impurity model, where flat part is absent ($T*\chi_s$. monotonously increases). I suggest the authors to cite the papers [22,25] (and possibly others) instead.

Indeed, if one scans the whole temperature range from the high-$T$ perturbative regime down to $T \rightarrow 0$, the quantity $T \chi_s(T)$ for the AIM we considered displays a non monotonous behavior with a rather broad maximum at about $T \leq \frac{U}{2}$. We agree, nonetheless, with the Referee, that our statement about a ``flat part'' of the quantity $T \chi_s(T)$ was rather imprecise and, in general, difficult to be quantified. For that reason, and also in the light of the observation made by the second Referee, in the revised manuscript we have dropped the qualitative statement mentioned above and have refined the corresponding discussion, which also benefited from the additional inclusion of a dedicated figure (showing the behavior of $T \chi_s(T)$ for the HA and the AIM) in the Appendix.

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